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CC Algebra I Lesson 31 Series & Sequences Number: _____ Name:__________________________ Date:___________________________ Aim: What are sequences? How can we write the formula for a given pattern? Do Now: Find the next three terms in each of the following patterns (a) (b) 3,6,9,12,... 1 (c) 1 1 1 1 1, , , , ,... 2 3 4 5 2,4,8,16,32,... EXPLORATION: Describing a Pattern Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use the figures to complete the table. Plot the points given by your completed table. Describe the pattern of the y-values. a. n =1 n=2 n=3 n=4 n=5 y 60 50 40 30 20 Number of stars, n 1 2 3 4 10 0 0 5 1 2 3 4 5n 1 2 3 4 5n Number of sides, y b. n =1 n=2 n=3 n=4 n=5 y 6 5 4 3 n Number of circles, y 1 2 3 4 5 2 1 0 0 A sequence is an ordered list of elements. It is a function whose set of inputs are x -values. A sequence is often shown as an ordered list of numbers, called the terms or elements of the sequence. Terms in a sequence are typically indexed by a subscript starting at 1. Example 1: Consider the sequence below. 2, 5, 8, 11, 14, 17, 20, 23… Sequences can be expressed in various ways. (a) The chart below is filled in based on the sequence above (b) Now create a chart using the subscript notation Term Number Term 1 2 Sequence Term a1 2 5 a2 3 8 a3 4 11 a4 5 14 a5 Term (c) Now create a chart using function notation Sequence Term Term a(n) Example 2: Consider the sequence below. If we represent this sequence with the letter a then complete the following: 4, 8, 16, 32, 64, 128, 256, 512 a1 __ ___ ___ ___ ____ ____ ____ (a) Find a (3) (b) Find a(1) a(7) (c) Find a (n) 16 (d) Find a 2 (e) Find a (1) a(3) (f) Find a (n) 128 Example 3: Consider the sequence defined by the formula a (n) 2n 1 . (a) Write out the first 5 elements of this sequence. (b) Graph the sequence on the grid shown below for 1 n 5. (c) Why shouldn't we connect the points plotted in a continuous straight line? (d) What is the 21st term of this sequence? Show how you arrived at your answer. Sequences can be defined by a classic function formula, like what we saw in Example #3, and they also can be defined recursively. A recursive formula is one where each term in the sequence depends on a term or term that came before it. Example 4: Consider the sequence of numbers given by the following definition: a1 = 3 and an an1 4 (a) Give a common sense interpretation for this recursive function rule. an 1 Represents the terms number (b) Find the first four terms. Example 5: Consider the sequence of numbers given by the following definition. a1 = 2 and an 5a n1 (a) Give a common sense interpretation for this recursive function rule. an 1 Represents the terms number (b) Find the first four terms. Recursive finds the patterns and can be used to find the next few terms. How would we find the nth term? To be continued…….. Practice Problems: 1) Consider the sequence below. If we represent this sequence with the letter a then do the following: 1, 7, 13, 19, 25, 31, 37, 43… __ __ __ ___ ___ ___ ___ ___ (a) Find a (5) (b) Find a1 a2 (c) Find a (1) a(3) (d) Find a(4) (e) Find a (n) 31 (f) Find a (n) 19 2) Consider the sequence below. If we represent this sequence with the letter a then do the following: 8, 17, 26, 35, 44, 53, 62, 71… __ ___ ___ ___ ___ ___ ___ ___ (a) Find a (3) (b) Find a1 a2 (c) Find a (2) a (3) (d) Find a(7) (e) Find a(n) 53 (f) Find a (n) 71