A geometric series is a sequence where each term is found by multiplying the previous term by a constant called the common ratio. The document defines the common ratio r and provides the general formula for calculating any term Tn in a geometric series. It also gives examples of (i) finding r and the general term for the series 2, 8, 32,... and (ii) finding r if T2=7 and T4=49. Finally, it solves (iii) finding the first term greater than 500 for the series 1, 4, 16,...
The document defines an arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. The nth term (Tn) of an arithmetic series is equal to the first term (a) plus (n-1) times the common difference (d). An example shows how to find the general term and other properties of a series given two terms. Specifically, if T3=9 and T7=21, the general term is Tn=3n-3 and the first term greater than 500 is T167=501.
The document discusses the Master Theorem for solving recurrence relations of the form T(n) = aT(n/b) + f(n). There are 3 cases depending on how f(n) compares to nlogba. 22 practice problems are given to apply the Master Theorem. For each, it is determined whether the Master Theorem can be used to solve the recurrence and express T(n), or if it does not apply.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document defines an arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. The nth term (Tn) of an arithmetic series is equal to the first term (a) plus (n-1) times the common difference (d). An example shows how to find the general term and other properties of a series given two terms. Specifically, if T3=9 and T7=21, the general term is Tn=3n-3 and the first term greater than 500 is T167=501.
The document discusses the Master Theorem for solving recurrence relations of the form T(n) = aT(n/b) + f(n). There are 3 cases depending on how f(n) compares to nlogba. 22 practice problems are given to apply the Master Theorem. For each, it is determined whether the Master Theorem can be used to solve the recurrence and express T(n), or if it does not apply.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
11X1 T14 01 definitions & arithmetic series (2011)Nigel Simmons
The document defines arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. It provides the general formula for an nth term in an arithmetic series as Tn = a + (n-1)d, where a is the first term and d is the common difference. As an example, it calculates the general term for a series where T3 = 9 and T7 = 21, finding the common difference is 3 and the first term is also 3, giving the general term as Tn = 3n - 3.
11 x1 t14 01 definitions & arithmetic series (2012)Nigel Simmons
The document defines arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. It provides the general formula for an nth term in an arithmetic series as Tn = a + (n-1)d, where a is the first term and d is the common difference. As an example, it calculates the general term for a series where T3 = 9 and T7 = 21, finding the common difference d = 3 and first term a = 3, giving the formula Tn = 3n - 3. It is then asked to calculate the 100th term T100 for this series.
11X1 T14 01 definitions & arithmetic series (2010)Nigel Simmons
The document defines arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. It provides the general formula for an nth term in an arithmetic series as Tn = a + (n-1)d, where a is the first term and d is the common difference. As an example, it calculates the general term for a series where T3 = 9 and T7 = 21, finding the common difference d = 3 and first term a = 3, giving the formula Tn = 3n - 3. It is then asked to calculate the 100th term T100 for this series.
The document defines a geometric series as a sequence of numbers where each subsequent term is found by multiplying the previous term by a constant ratio. The constant ratio is called the common ratio and is represented by r. Examples are provided to demonstrate calculating terms of a geometric series given the first term and common ratio. The concept of finding the first term greater than a given value is also illustrated.
11 x1 t14 01 definitions & arithmetic series (2013)Nigel Simmons
This document defines arithmetic series and provides examples of solving problems related to arithmetic series. It begins by defining an arithmetic series as a sequence where each term is found by adding a constant amount to the previous term. It then gives the general formula for finding the nth term and provides examples of using the formula to find specific terms and solve other problems such as determining the first term greater than a given value.
Fourier series can be used to decompose periodic functions into simpler trigonometric components. A periodic function can be represented as the sum of an infinite series of sines and cosines with frequencies that are integer multiples of a fundamental frequency. This decomposition allows periodic waveforms to be analyzed and approximated by truncating the series to include only the first few terms. The sine and cosine functions form an orthogonal basis set for periodic functions, which means the Fourier series representation is unique. An example shows how a square wave can be represented by its Fourier series expansion using only sine terms.
The document discusses sequences and provides examples of arithmetic and geometric sequences. It defines key terms like sequence, arithmetic sequence, common difference, geometric sequence, and common ratio. Formulas are given for finding the nth term in an arithmetic sequence and a geometric sequence. Examples are shown for calculating the common difference and finding a specific term in an arithmetic sequence.
This document defines sequences and provides information about arithmetic and geometric sequences. It discusses the recursive and implicit definitions of sequences. For arithmetic sequences, it explains how to find the common difference and the nth term using the common difference and the first term. For geometric sequences, it similarly explains how to find the common ratio and the nth term using the common ratio and first term.
The document defines sequences and provides information about arithmetic and geometric sequences. It explains that:
- An arithmetic sequence follows a pattern where each term is generated by adding a common difference to the previous term. The common difference is the slope of the linear equation that generates the terms.
- The nth term in an arithmetic sequence can be calculated as: tn = a + (n - 1)d, where a is the first term, n is the position of the term, and d is the common difference.
- A geometric sequence follows a pattern where each term is generated by multiplying the previous term by a common ratio. The common ratio is the base of the exponential equation that generates the terms.
Introduction to sequences, arithmetic, geometric and others. Recursively and implicitly definitions. Using the graphing calculator find the value of any term in a sequence.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The document defines sequences and their patterns, including arithmetic and geometric sequences. It provides the recursive and implicit definitions of each, explaining how to find the common difference or ratio, and the nth term in the sequences. Examples are given for finding the common difference and the nth term in an arithmetic sequence, and the implicit definition and nth term for a geometric sequence.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
11X1 T14 01 definitions & arithmetic series (2011)Nigel Simmons
The document defines arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. It provides the general formula for an nth term in an arithmetic series as Tn = a + (n-1)d, where a is the first term and d is the common difference. As an example, it calculates the general term for a series where T3 = 9 and T7 = 21, finding the common difference is 3 and the first term is also 3, giving the general term as Tn = 3n - 3.
11 x1 t14 01 definitions & arithmetic series (2012)Nigel Simmons
The document defines arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. It provides the general formula for an nth term in an arithmetic series as Tn = a + (n-1)d, where a is the first term and d is the common difference. As an example, it calculates the general term for a series where T3 = 9 and T7 = 21, finding the common difference d = 3 and first term a = 3, giving the formula Tn = 3n - 3. It is then asked to calculate the 100th term T100 for this series.
11X1 T14 01 definitions & arithmetic series (2010)Nigel Simmons
The document defines arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. It provides the general formula for an nth term in an arithmetic series as Tn = a + (n-1)d, where a is the first term and d is the common difference. As an example, it calculates the general term for a series where T3 = 9 and T7 = 21, finding the common difference d = 3 and first term a = 3, giving the formula Tn = 3n - 3. It is then asked to calculate the 100th term T100 for this series.
The document defines a geometric series as a sequence of numbers where each subsequent term is found by multiplying the previous term by a constant ratio. The constant ratio is called the common ratio and is represented by r. Examples are provided to demonstrate calculating terms of a geometric series given the first term and common ratio. The concept of finding the first term greater than a given value is also illustrated.
11 x1 t14 01 definitions & arithmetic series (2013)Nigel Simmons
This document defines arithmetic series and provides examples of solving problems related to arithmetic series. It begins by defining an arithmetic series as a sequence where each term is found by adding a constant amount to the previous term. It then gives the general formula for finding the nth term and provides examples of using the formula to find specific terms and solve other problems such as determining the first term greater than a given value.
Fourier series can be used to decompose periodic functions into simpler trigonometric components. A periodic function can be represented as the sum of an infinite series of sines and cosines with frequencies that are integer multiples of a fundamental frequency. This decomposition allows periodic waveforms to be analyzed and approximated by truncating the series to include only the first few terms. The sine and cosine functions form an orthogonal basis set for periodic functions, which means the Fourier series representation is unique. An example shows how a square wave can be represented by its Fourier series expansion using only sine terms.
The document discusses sequences and provides examples of arithmetic and geometric sequences. It defines key terms like sequence, arithmetic sequence, common difference, geometric sequence, and common ratio. Formulas are given for finding the nth term in an arithmetic sequence and a geometric sequence. Examples are shown for calculating the common difference and finding a specific term in an arithmetic sequence.
This document defines sequences and provides information about arithmetic and geometric sequences. It discusses the recursive and implicit definitions of sequences. For arithmetic sequences, it explains how to find the common difference and the nth term using the common difference and the first term. For geometric sequences, it similarly explains how to find the common ratio and the nth term using the common ratio and first term.
The document defines sequences and provides information about arithmetic and geometric sequences. It explains that:
- An arithmetic sequence follows a pattern where each term is generated by adding a common difference to the previous term. The common difference is the slope of the linear equation that generates the terms.
- The nth term in an arithmetic sequence can be calculated as: tn = a + (n - 1)d, where a is the first term, n is the position of the term, and d is the common difference.
- A geometric sequence follows a pattern where each term is generated by multiplying the previous term by a common ratio. The common ratio is the base of the exponential equation that generates the terms.
Introduction to sequences, arithmetic, geometric and others. Recursively and implicitly definitions. Using the graphing calculator find the value of any term in a sequence.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The document defines sequences and their patterns, including arithmetic and geometric sequences. It provides the recursive and implicit definitions of each, explaining how to find the common difference or ratio, and the nth term in the sequences. Examples are given for finding the common difference and the nth term in an arithmetic sequence, and the implicit definition and nth term for a geometric sequence.
Similar to 11X1 T14 02 geometric series (2010) (13)
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
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Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
2. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
3. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
4. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T
r 2
a
5. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T
r 2
a
T
3
T2
6. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T
r 2
a
T
3
T2
Tn
r
Tn1
7. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a
T
3
T2
Tn
r
Tn1
8. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T
3
T2
Tn
r
Tn1
9. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T3 T3 ar 2
T2
Tn
r
Tn1
10. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T3 T3 ar 2
T2 Tn ar n1
Tn
r
Tn1
11. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T3 T3 ar 2
T2 Tn ar n1
Tn
r
Tn1 e.g.i Find r and the general term of 2, 8, 32,
12. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T3 T3 ar 2
T2 Tn ar n1
Tn
r
Tn1 e.g.i Find r and the general term of 2, 8, 32,
a 2, r 4
13. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T3 T3 ar 2
T2 Tn ar n1
Tn
r
Tn1 e.g.i Find r and the general term of 2, 8, 32,
Tn ar n1 a 2, r 4
14. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T3 T3 ar 2
T2 Tn ar n1
Tn
r
Tn1 e.g.i Find r and the general term of 2, 8, 32,
Tn ar n1 a 2, r 4
24
n1
15. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T3 T3 ar 2
T2 Tn ar n1
Tn
r
Tn1 e.g.i Find r and the general term of 2, 8, 32,
Tn ar n1 a 2, r 4
24
n1
22
2 n 1
22
2 n2
16. Geometric Series
An geometric series is a sequence of numbers in which each term after
the first is found by multiplying a constant amount to the previous
term.
The constant amount is called the common ratio, symbolised, r.
T T1 a
r 2
a T2 ar
T3 T3 ar 2
T2 Tn ar n1
Tn
r
Tn1 e.g.i Find r and the general term of 2, 8, 32,
Tn ar n1 a 2, r 4
24
n1
22
2 n 1
Tn 22 n1
22
2 n2
19. ii If T2 7 and T4 49, find r
ar 7
ar 3 49
20. ii If T2 7 and T4 49, find r
ar 7
ar 3 49
r2 7
21. ii If T2 7 and T4 49, find r
ar 7
ar 3 49
r2 7
r 7
22. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
ar 3 49
r2 7
r 7
23. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
ar 3 49 a 1, r 4
r2 7
r 7
24. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
Tn 14
n1
ar 3 49 a 1, r 4
r2 7
r 7
25. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
Tn 14
n1
ar 3 49 a 1, r 4
r2 7 Tn 500
r 7
26. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
Tn 14
n1
ar 3 49 a 1, r 4
r2 7 Tn 500
r 7 4n1 500
27. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
Tn 14
n1
ar 3 49 a 1, r 4
r2 7 Tn 500
r 7 4n1 500
log 4n1 log 500
28. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
Tn 14
n1
ar 3 49 a 1, r 4
r2 7 Tn 500
r 7 4n1 500
log 4n1 log 500
n 1log 4 log 500
29. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
Tn 14
n1
ar 3 49 a 1, r 4
r2 7 Tn 500
r 7 4n1 500
log 4n1 log 500
n 1log 4 log 500
n 1 4.48
n 5.48
30. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
Tn 14
n1
ar 3 49 a 1, r 4
r2 7 Tn 500
r 7 4n1 500
log 4n1 log 500
n 1log 4 log 500
n 1 4.48
n 5.48
T6 1024, is the first term 500
31. ii If T2 7 and T4 49, find r (iii) find the first term of 1, 4, 16, … to
ar 7 be greater than 500.
Tn 14
n1
ar 3 49 a 1, r 4
r2 7 Tn 500
r 7 4n1 500
log 4n1 log 500
n 1log 4 log 500
n 1 4.48
n 5.48
T6 1024, is the first term 500
Exercise 6E; 1be, 2cf, 3ad, 5ac, 6c, 8bd, 9ac, 10ac, 15, 17,
18ab, 20a