Arithmetic Sequence
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The document discusses arithmetic sequences and how to find terms in arithmetic sequences. It defines an arithmetic sequence as a sequence where each term is obtained by adding a constant difference to the preceding term. It provides the formula for finding the nth term in an arithmetic sequence and works through examples of identifying arithmetic sequences and finding specified terms.
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Arithmetic Sequence
The document defines sequences and series, and discusses arithmetic sequences in particular. An arithmetic sequence is a sequence where each term after the first is obtained by adding a fixed number, called the common difference, to the preceding term. The document provides examples of finding the common difference using the formulas d=an+1-an and d=an-a1. It also gives examples of finding specific terms of an arithmetic sequence given information like the first term, common difference, and nth term.
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This document discusses arithmetic and geometric sequences. It defines arithmetic sequences as having a constant difference between consecutive terms, called the common difference. Geometric sequences have a constant ratio between consecutive terms, called the common ratio. Formulas are provided for finding the nth term of an arithmetic sequence and a geometric sequence based on the initial term and common difference or ratio. Examples are given of identifying the type of sequence and calculating terms. The document also discusses notation, formal definitions, and graphing sequences.
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1. The document instructs readers to fold a square piece of paper into quarters and record the resulting areas in a table after each fold.
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This document provides information about arithmetic sequences, including:
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- Formulas are provided for finding individual terms and the sum of terms in an arithmetic sequence based on the first term, common difference, and number of terms.
- An example is given of Carl Friedrich Gauss efficiently solving the problem of adding all numbers from 1 to 100, showing the relationship between arithmetic sequences and pairs of numbers that sum to a constant.
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- Formulas are provided for finding individual terms and the sum of terms in an arithmetic sequence based on the first term, common difference, and number of terms.
- An example is given of Carl Friedrich Gauss efficiently solving the problem of adding all numbers from 1 to 100, showing the relationship between arithmetic sequences and pairs of numbers that sum to a constant.
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2) Inverse variation means that as one variable increases, the other decreases, maintaining a constant product. The graph of an inverse variation is a hyperbola.
3) Examples show inverse variations between variables like pressure and volume of a gas, or jobs completed and number of workers.
4) Quiz questions test understanding of direct, inverse, and their equation representations.
Philosophy of education
This document contains a philosophy of education report submitted by Pilar M. Castillo of Muntinlupa National High School Annex. It includes tongue twisters, riddles about philosophers such as Plato, Aristotle, Confucius, and Immanuel Kant. It also lists philosophies like naturalism, idealism, realism, experimentalism, existentialism, socialism, and communism along with their key knowledge, values, and beliefs. At the end is a quote by Henry David Thoreau about what it means to be a philosopher.
Alamat ng Ampalaya
This document appears to be a nonsensical collection of words and symbols with no clear meaning or message that can be summarized in 3 sentences or less. It does not contain any essential information that could be extracted to create a concise and accurate summary.
System of linear equations and their solution
1. The document presents examples of solving addition and subtraction problems with positive and negative numbers. It also gives examples of finding two numbers whose sum and difference satisfy given values.
2. It defines a system of linear equations as two equations in two variables of the form Ax + By = C and Dx + Ey = F. It explains that an ordered pair (x,y) is a solution if it satisfies both equations.
3. Examples are given of determining if an ordered pair is a solution to a given system of linear equations by substituting the values into each equation and checking if both equations are satisfied. An activity asks students to identify if ordered pairs are solutions to several systems.
Betweeness & collinearity
This document discusses the concept of betweenness for collinear points on a number line. It provides examples of determining which point is between two other points given distances between points. It also includes practice problems identifying betweenness based on given segment measures between collinear points.
Pascal Triangle
This document contains information from a mathematics class. It discusses expanding binomial expressions using Pascal's triangle, worked examples of finding specific terms in expansions, and an activity for students to practice expansions. It also presents the "monkey donkey paradox" word problem and asks students to find the number of cupcakes eaten on subsequent days if the pattern continues.
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Arithmetic Sequence
- 1. Presented by: Joseph C. Nilo
- 2. Practice Exercise: Find the next five terms of the following sequence: • 1, 3, 5, 7, … • 2, 4, 6, 8, … • 1, 5, 9, 8, … • -7, -2, 3, 8, … • 20, 11, 2, -7, …
- 3. Problem of the day!!! • Find the common difference and the requires term in the following sequence • 3, 6, 9, … a8 • -5, -10, -15, … a20 • 8, -1, -10, -19, … a18
- 4. Arithmetic Sequence • A sequence in which each term is obtained by adding a constant d to preceding term. The constant number d is called the common difference. • The nth term of an arithmetic sequence is defined as an = a1 + (n – 1)d
- 5. a n = a 1 + (n – 1)d where – a n = the last term – a 1 = the first term – n = the number of term – d = the common difference
- 7. • State whether or not each of the following is an arithmetic progression. If it is, give the common difference. 5, 10, 15, … -5, 6, 10, … 4, 5, 6, 7, … 6, 20, 30, … ½, ¼, 1/6, … d = 5 not d = 1 not not
- 8. • Find the indicated term of the arithmetic sequence. 1. a1 = 10, d = 6, find a7 2. a1 = 12, d = 2, find a10 3. a1 = -5, d = -7, a12 = ? 4. a1 = -20, d = -4, a8 = ? 5. a1 = 0.5, d = 1.5, find a28 46 30 -82 -48 41
- 10. • Find the indicated term of the arithmetic sequence. 1. 12, 20, 28, … find a25 2. -5, -13, -21, … find a43 3. 2.5, 7, 11.5, … find a9 4. 15, 17, 19, … find a35 5. -3.8, -6.2, -8.6, … find a20