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ALL WORK (NEATLY ORGANIZED) IN A NOTEBOOK Name: ______________________________________________________________ Date: _______________________ Period: ______ Series Recall: A sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series. In an arithmetic sequence: To find the sum of a certain number of terms of a arithmetic sequence: *given on the regents exam reference sheet where Sn is the sum of n terms (nth partial sum), a1 is the first term, an is the nth term. To find the arithmetic series, the formula for an must be used first (same formula as yesterday) Model Solution Ex: Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ... To use the sum formula, an needs to be found first: a1 = 4, n = 20, d = 2 a20 = 4 + (20 – 1)(2) a20 = 4 + 19(2) a20 = 4 + 38 a20 = 42 Now the sum formula can be used. n = 20, a1 = 4, an = a20 = 42 Exercises: O. Answer each question 48.) Find the sum of the first 30 terms of 5, 9, 13, 17, ... 49.) Determine the sum of the first 17 terms of the arithmetic sequence whose first 4 terms are -15, -9, -3, 3 50.) Determine the sum of the first 8 terms of the arithmetic sequence whose first 4 terms are 8, 11, 14, 17 51.) Find the sum of the arithmetic series 3, 6, 9, .... ,99 52.) Determine the sum of 22, 16, 10, … , -80 ALL WORK (NEATLY ORGANIZED) IN A NOTEBOOK In a geometric sequence: To find the sum of a certain number of terms of a geometric sequence: *given on the regents exam reference sheet where Sn is the sum of n terms (nth partial sum), a1 is the first term, r is the common ratio. Model Solution Ex: Find the sum of the first 8 terms of the sequence -5, 15, -45, 135, ... a1 = -5, r = -3, n = 8 S8 = S8 = S8 = S8 = S8 = 8200 Exercises: P. Answer each question 53.) Determine the sum of the first 15 terms of the geometric sequence 1, 2, 4, 8, …. 54.) Determine the sum of the first 11 terms of the geometric sequence 2, -6, 18, -54, …. 55.) Determine the sum of the first 6 terms of the geometric sequence 1000, 200, 40, 8, …. 56.) Determine the sum of the first 9 terms of the geometric sequence 1, 6, 36, 216, …. ALL WORK (NEATLY ORGANIZED) IN A NOTEBOOK Recursion: Recursion is the process of choosing a starting term and repeatedly applying the same process to each term to arrive at the following term. Recursion requires that you know the value of the term immediately before the term you are trying to find. A recursive formula always has two parts: 1. the starting value for a1. 2. the recursion equation for an as a function of an-1 (the term before it.) Consider the sequence 2, 5, 8, 11, ... The recursive formula is This recursion formula could be used to identify many more terms in the series Consider the sequence 3, 12, 48, 192, ... The recursive formula is This recursion formula could be used to identify many more terms in the series The recursion formulas do not always have to be a simple arithmetic or geometric series. As long as it follows a pattern, it can be worked with Model Solution: Ex: Given the recursive formula, write the first 4 terms of the sequence: a1 = -4 a2 = -4 + 5 = 1 a3 = 1 + 5 = 6 a4 = 6 + 5 = 11 -4, 1, 6, 11 Exercises: Q. Answer each question 57.) Find the first 4 terms of the sequence 58.) Write the first 5 terms of the sequence 59.) Write a recursive formula for the sequence 9, -18, 36, -72, ... 60.) Write a recursive formula for the sequence 61.) Write a recursive formula for the sequence 5, 11, 23, 47, 95, …