Chapter 1 - Arithmetic and Geometric Sequences PDF
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Uploaded by FortuitousChrysanthemum
Universiti Malaya
Sir. Akmarul
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Summary
This document covers arithmetic and geometric sequences. It includes objectives, definitions of sequences, examples, formulas, and exercises. The document also provides notation, explanations, and example questions.
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– Chapter 1 – Arithmetic and Geometric Sequences Objectives: 1. Identify arithmetic and geometric sequences 2. Find formulas for the nth term Sir. Akmarul What is a Sequence? A set of numbers, called terms, arranged in a parti...
– Chapter 1 – Arithmetic and Geometric Sequences Objectives: 1. Identify arithmetic and geometric sequences 2. Find formulas for the nth term Sir. Akmarul What is a Sequence? A set of numbers, called terms, arranged in a particular order. Two simplest types: Arithmetic Geometric Arithmetic Sequences Difference between consecutive terms is constant. Called the “common difference.” Examples: 2, 6, 10, 14, 18, … diff. = 4 17, 10, 3, -4, -11, -18, … a, a+d, a+2d, a+3d, a+4d, … diff. = -7 diff. = d Geometric Sequences Ratio of consecutive terms is constant. Called the “common ratio.” Examples: 1, 3, 9, 27, 81, … ratio= 3 64, -32, 16, -8, 4, … a, ar, ar , ar , ar , … ratio = -1/2 2 3 4 ratio = r You Try! Identify the type of sequence and the common difference or ratio. 5, 10, 20, 40, … 6, 1, -4, -9, … Notation 1 term: t , 2 term: t , n term: t st 1 nd 2 th n Some sequences can be defined by rules or formulas. Ex: t n = n2 + 1 t1 = 12 + 1 = 2 t2 = 22 + 1 = 5, and so on Arithmetic Formulas nth term 0th term add the (work difference backwards n times. to find) tn = t0 + dn Geometric Formulas nth term 0th term multiply by the ratio n times tn = t0 ∙ r n Formal Definition “A function whose domain is the set of positive integers.” For example: The sequence tn = 4n – 2 - can be thought of as - The function t(n) = 4n – 2 (where n is a + integer) Graphing Sequences Write terms as ordered pairs and plot. Ex: 1, 4, 7, 10, … has points (1, 1), (2, 4), (3, 7), (4, 10) Notice n (the term number) is the x! Arithmetic – points lie on a line Geometric – points lie on an exponential curve Example 1: Find formula for nth term of 3, 5, 7… Sketch the graph. Example 2: Find formula for nth term of 3, 4.5, 6.75… Sketch the graph. You Try! Find formula for nth term of 15, 7, -1, -9, … 100, -50, 25, -12.5, … Example 3: In a geometric sequence, t 3 = 12 and t6 = 96. Find t11. Example 4: In an arithmetic sequence t 2 = 2 and t5 = 16. Find t10. You Try! In a geometric sequence t 2 = 2 and t5 = 16. Find t10. The End.
