Notes 8-4
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This document discusses finding the formula for the sum of the first n terms of a geometric series. It states that the formula for the sum of the first n terms of the geometric series 1 + 3 + 9 + ... is Sn = (3^n - 1)/2. It then asks to use this formula to determine the sum of the first 10 terms of the given series.
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6.sequences and series Further Mathematics Zimbabwe Zimsec Cambridge
6.sequences and series Further Mathematics Zimbabwe Zimsec Cambridge
More from Jimbo Lamb
Geometry Section 1-5
The document defines and provides examples of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It defines each term and provides examples of identifying angle pairs that satisfy each relationship. It also includes examples of using properties of these angle relationships to solve problems, such as finding missing angle measures.
Geometry Section 1-4
The document defines various terms related to angle measure including ray, angle, vertex, acute angle, obtuse angle, and angle bisector. It then provides examples measuring angles in a figure and solving an equation involving angle measures.
Geometry Section 1-3
This document discusses finding points and midpoints on line segments. It defines midpoint as the point halfway between two endpoints and provides the formula to calculate it. Several examples are given to demonstrate how to find the midpoint of a segment, locate a point at a fractional distance from one endpoint, and find a point where the ratio of distances from the endpoints is a given ratio. The key concepts covered are calculating midpoints using averages of x- and y-coordinates, and setting up and solving equations to locate interior points using fractional or ratio distances along a segment.
Geometry Section 1-2
This document defines key vocabulary terms related to line segments and distance. It defines a line segment as a portion of a line distinguished by endpoints, and defines betweenness of points and the term "between." It also defines congruent segments, constructions, distance, and irrational numbers. Several examples are provided to demonstrate calculating distances between points on a number line, using a ruler to measure segments, and applying the Pythagorean theorem.
Geometry Section 1-2
This document defines key vocabulary terms related to line segments and distance, including line segment, betweenness of points, congruent segments, and distance formula. It provides examples of calculating distances between points on number lines and using the Pythagorean theorem to find distances between points graphed on a coordinate plane. Examples include measuring line segments with a ruler, finding distances by adding or subtracting measures, and applying the distance formula and Pythagorean theorem to solve for unknown distances.
Geometry Section 1-1
This document introduces basic geometry concepts such as points, lines, planes, and their intersections. It defines a point as having no size or shape, a line as an infinite set of collinear points, and a plane as a flat surface extending indefinitely. Examples demonstrate identifying geometric shapes from real-world objects and graphing points and lines on a coordinate plane. The summary defines key terms and provides examples of geometric concepts and relationships.
Algebra 2 Section 5-3
The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.
Algebra 2 Section 5-2
The document discusses composition of functions. It defines composition of functions as using the output of one function as the input of another. It provides an example of composing two functions f and g, showing the steps of evaluating f(g(x)) and g(f(x)) at different values of x. Another example is given with two functions defined by sets of ordered pairs, finding the compositions f∘g and g∘f by evaluating them at different inputs and stating their domains and ranges.
Algebra 2 Section 5-1
The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, and division. Definitions of each operation are provided, along with examples of applying the operations to specific functions. Addition of functions involves adding the outputs of each function, subtraction involves subtracting the outputs, multiplication involves multiplying the outputs, and division involves dividing the outputs given the denominator function is not equal to 0. Several examples are worked through applying the different operations to functions like f(x)=2x and g(x)=-x+5. The examples also demonstrate evaluating composite functions and restricting domains as needed.
Algebra 2 Section 4-9
The document discusses determining the number and type of roots of polynomial equations. It states that every polynomial with degree greater than zero has at least one root in the set of complex numbers according to the Fundamental Theorem of Algebra. Descartes' Rule of Signs is introduced, which relates the number of changes in sign of a polynomial's terms to its possible positive and negative real roots. An example problem is worked through applying these concepts to determine the possible number of positive, negative, and imaginary roots.
Algebra 2 Section 4-8
This document discusses synthetic division and the remainder and factor theorems. It provides examples of using synthetic division to evaluate functions, determine the number of terms in a sequence, and factor polynomials. The key steps of synthetic division are shown, along with checking the remainder and determining common factors. Three examples are worked through to demonstrate these concepts.
Algebra 2 Section 4-6
This document discusses solving polynomial equations by factoring polynomials. It begins with essential questions and vocabulary about factoring polynomials and solving polynomial equations by factoring. It then provides the number of terms in a polynomial and the corresponding factoring technique that can be used. Examples of factoring various polynomials are also provided. The document aims to teach students how to factor polynomials and solve polynomial equations by factoring.
Geometry Section 6-6
The document defines key terms and theorems related to trapezoids and kites. It provides definitions for trapezoid, bases, legs of a trapezoid, base angles, isosceles trapezoid, midsegment of a trapezoid, and kite. It also lists theorems about properties of isosceles trapezoids and kites. Two examples problems are included, one finding measures of an isosceles trapezoid and another showing a quadrilateral is a trapezoid.
Geometry Section 6-5
The document discusses rhombi and squares. It defines a rhombus as a parallelogram with four congruent sides and gives its properties. A square is defined as a parallelogram with four right angles and four congruent sides. The document provides theorems for identifying rhombi and squares. It then gives examples of using the properties and theorems to determine if a shape is a rhombus, rectangle, or square.
Geometry Section 6-4
The document discusses properties of rectangles. A rectangle is defined as a parallelogram with four right angles. The key properties are that opposite sides are parallel and congruent, opposite angles are congruent, and consecutive angles are supplementary. The diagonals of a rectangle bisect each other and are congruent. Theorems are presented regarding the diagonals of rectangles. Examples apply the properties of rectangles to find missing side lengths, angles, and diagonals. One example uses the distance formula and slope to determine if a quadrilateral is a rectangle.
Geometry Section 6-3
The document discusses properties of parallelograms and provides examples of determining if a quadrilateral is a parallelogram. It defines four theorems for identifying parallelograms based on opposite sides, opposite angles, bisecting diagonals, and parallel/congruent sides. Examples solve systems of equations to find values of variables such that the quadrilaterals satisfy parallelogram properties. One example uses slopes of side segments to show a quadrilateral is a parallelogram due to parallel opposite sides.
Geometry Section 6-2
The document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with two pairs of parallel sides. It then lists several properties of parallelograms: opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and if one angle is a right angle all angles are right angles. It also discusses properties of diagonals in parallelograms, including that diagonals bisect each other and divide the parallelogram into two congruent triangles. Several examples demonstrate using these properties to solve problems about parallelograms.
Geometry Section 6-1
The document summarizes key concepts about polygons, including:
- The sum of the interior angles of a polygon with n sides is (n-2)180 degrees.
- The sum of the exterior angles of a polygon is 360 degrees.
- Examples are provided to demonstrate calculating sums of interior/exterior angles and finding missing angle measures using angle sums.
- Regular polygons are defined by their number of sides.
Algebra 2 Section 4-5
The document discusses analyzing graphs of polynomial functions. It provides examples of locating real zeros of polynomials using the location principle and estimating relative maxima and minima. Example 1 analyzes the polynomial f(x) = x^4 - x^3 - 4x^2 + 1 and locates its real zeros between consecutive integer values. Example 2 graphs the polynomial f(x) = x^3 - 3x^2 + 5 and estimates the x-coordinates of relative maxima and minima.
Algebra 2 Section 4-4
This document discusses polynomial functions. It defines key terms like polynomial in one variable, leading coefficient, and polynomial function. It provides examples of power functions of varying degrees like quadratic, cubic, quartic and quintic functions. The document also includes examples of evaluating polynomial functions, finding degrees and leading coefficients, graphing polynomial functions from tables of values, and describing properties of graphs.
More from Jimbo Lamb (20)
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Paris 2024 History-making Matildas team selected for Olympic Games.pdf
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Psaroudakis: Family and Football – The Psaroudakis Success Story
Psaroudakis, a name that resonates with football fans around the globe, is a testament to the powerful synergy between familial support and individual passion. Born on March 10, 1992, in the historic city of Heraklion, Crete, Psaroudakis’ journey to international football stardom is a compelling narrative of dedication, perseverance, and unwavering family support. His story not only highlights his athletic prowess but also underscores the crucial role his family played in shaping his career and character.
Psaroudakis’ early life in Heraklion was deeply influenced by a supportive and nurturing family environment. His father, a former semi-professional footballer, recognized Psaroudakis’ potential from an early age. Acting as his first coach, his father’s guidance was instrumental in igniting Psaroudakis’ passion for football. This paternal influence instilled in him a strong work ethic and fundamental skills that would become the foundation of his future success. His mother, a dedicated homemaker, provided a stable and nurturing environment, ensuring that Psaroudakis could pursue his dreams without any hindrances.
From a young age, Psaroudakis showed an innate talent for football. Growing up in Heraklion, he spent countless hours playing football in local parks and streets with friends and family. His natural ability was evident even in these informal settings, and his enthusiasm for the game was infectious. By the age of five, Psaroudakis had joined a local youth football club, where his skills began to flourish. His father’s role as his first coach during these formative years was crucial, as he emphasized not only technical skills but also the importance of discipline and teamwork.
The transition from playing in local parks to joining a structured football environment marked a significant step in Psaroudakis’ journey. At the age of ten, he joined the youth academy of OFI Crete, one of Greece’s most esteemed football clubs. This move marked the beginning of a more rigorous and professional approach to his training. The academy environment was demanding, focusing on honing technical abilities and instilling values of sportsmanship and dedication. Psaroudakis’ dedication to his craft was evident as he quickly rose through the ranks, becoming a standout player in the youth teams.
The support of Psaroudakis’ family was unwavering during this critical period. His father continued to be a source of guidance and mentorship, while his mother ensured that he had everything he needed to succeed. Their collective efforts created a balanced environment where Psaroudakis could focus entirely on his development as a footballer. This familial support was not just about providing the basics; it was about creating an environment where Psaroudakis felt encouraged and motivated to pursue his dreams relentlessly.
As Psaroudakis transitioned from the youth academy to professional football, the challenges became more significant.
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Hesan Soufi's Legacy: Inspiring the Next Generation
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Euro 2024 Predictions - Group Stage Outcomes
Here are our Euro 2024 predictions for the group stages
Will England make it through the group stages?, Will Germany use the home advantage to full effect?
Follow our progress, see how many we get right
If you want to join in let us know before the first game kick off and we can invite you to our private league
or join in with our friends at DeeperThanBlue
https://www.linkedin.com/posts/activity-7204868572995538944-qejG
https://www.selectdistinct.co.uk/2024/06/13/euro-2024-match-predictions/
#EURO2024 #Germany2024 #England #EURO2024predictions
The most promising young Players to watch during Euro 2024.docx
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Euro 2024 Belgium's Rebirth the New Generation Match the Golden Era.docx
The Golden Group is over. Can a new group step up? Two years ago, Kevin De Bruyne plunged Belgium’s Euro 2024 plans into disorder when he claimed the team was “too old” to win in an interview with The Protector. That Belgian squad had 10 players over 30 and the maximum average age of any Euro Cup 2024 team at the competition. A group-stage exit and just one goal at the World Cup put Belgium on course for a restructure.
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Belgium vs Romania Tickets | Euro Cup Tickets | Euro Cup Final Tickets
Coach Domenico Tedesco has managed a tactical shakeup and a regular exit for some of the oldest players. Experienced bests remain, not least the 37-year-old Jan Vertonghen in defense, the 32-year-old De Bruyne himself in midfield, and 31-year-old Romelu Lukaku up visible.
Still, younger actors like De Bruyne’s Manchester City partner Jeremy Doku bring fresh vitality to the team. Euro Cup Germany Qualifying unbeaten with just four goals allowed from eight games was a welcome sign of accomplishment back on track under Tedesco.
The only other squad in Group E besides Belgium to UEFA Euro 2024 qualify unbeaten, Romania was awestruck by winning a group that also checked Switzerland and Israel. Still, Euro 2024 will test a squad sorely lacking in top-level skill.
Euro 2024: Belgium's Transition from Golden Generation to New Hope
Tottenham guardian Vlad Dragusin is the only Euro Cup 2024 squad member singing regularly for one of Europe’s top clubs this flavor. He even played only nine Premier League games since adoption in January. Goalkeeper Horatiu Moldovan is a stoppage at Atletico Madrid.
There’s a link to the beauty days of Romanian soccer with midfielder Ianis Hagi, son of Gheorghe Hagi, who assisted the team to the rounds of the 1994 World Cup and Euro 2000.
We are only a combine of days away from the UEFA Euro 2024 curtain raiser. The 24 squads are winding up their provisions and getting ready to give it their all to life the wanted Euro Cup Final trophy on July 14. Spread across six clusters, the first hurdle in the knockout phase will be the plump of 16.
Euro fans worldwide can book Euro 2024 Tickets from our online platform, www.eticketing.co. Fans can book Euro Cup 2024 Tickets on our website at discounted prices.
Germany and Scotland will take things off before we get into overdrive in two weeks. Meanwhile, Belgium will be longing to bounce back after a horrendous 2022 FIFA World Cup movement, which ended in the group stage.
Belgium vs Romania Tickets | Euro Cup 2024 Tickets | Euro Cup Tickets | Euro Cup Final Tickets
Roberto Martinez completed the way for Domenico Tedesco, who has overseen a compact start to his tenure. The 38-year-old will be assured heading into the group stage
一比一原版(Curtin毕业证)科廷大学毕业证如何办理
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（详情请加一下 文凭顾问+微信:95270640）欢迎咨询！将要跨出门槛的一刹那山娃突然抱紧父亲的大腿嚎啕大哭父亲吃力地掰开山娃的手哽咽道山娃好好听话好好念书到了暑假爸也接你进城爸的城好远好大好美山娃于是天天扳着手指算计着读书也格外刻苦无奈时间总过得太慢太慢每次父亲往家打电话山娃总抢着接听一个劲地提醒父亲别忘了正月说的话电话那头总会传来父亲嘿嘿的笑连连说记得记得但别忘了拿奖状进城啊考试一结束山娃就迫不及待地给父亲挂电话：爸我拿奖了三好学生接我进城吧父亲果不
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Notes 8-4
- 1. Section 8-4 Geometric Series
- 2. Warm-up 1. Find a formula for the sum Sn of the first n terms of the geometric series 1 + 3 + 9 + ... 2. Use this formula to determine the sum of the first 10 terms of the series.
- 3. Warm-up 1. Find a formula for the sum Sn of the first n terms of the geometric series 1 + 3 + 9 + ... n 3 −1 Sn = 2 2. Use this formula to determine the sum of the first 10 terms of the series.