1052 CHAPTER 11 Further Topics in Algebra EXAMPLE 1 Finding the Common Difference Determine the common difference d for the arithmetic sequence -9, -7, -5, -3, -1, c . SOLUTION We find d by choosing any two adjacent terms and subtracting the first from the second. Choosing -7 and -5 gives the following. d = an+1 - an Common difference d d = -5 - 1-72 Let an+1 = -5 and an = -7. d = 2 Subtract. Choosing -9 and -7 would give d = -7 - 1-92 = 2, the same result. S Now Try Exercise 11. If a1 is the first term of an arithmetic sequence and d is the common difference, then the terms of the sequence are given as follows. a1 = a1 a2 = a1 + d a3 = a2 + d = a1 + d + d = a1 + 2d a4 = a3 + d = a1 + 2d + d = a1 + 3d a5 = a1 + 4d a6 = a1 + 5d By this pattern, an = a1 + 1n - 12d. EXAMPLE 2 FindingTerms Given a 1 and d Find the first five terms of each arithmetic sequence. (a) The first term is 7, and the common difference is -3. (b) a1 = -12, d = 5 SOLUTION (a) a1 = 7 Start with a1 = 7. a2 = 7 + 1-32 = 4 Add d = -3. a3 = 4 + 1-32 = 1 Add -3. a4 = 1 + 1-32 = -2 Add -3. a5 = -2 + 1-32 = -5 Add -3. (b) a1 = -12 Start with a1 = -12. a2 = -12 + 5 = -7 Add d = 5. a3 = -7 + 5 = -2 Add 5. a4 = -2 + 5 = 3 Add 5. a5 = 3 + 5 = 8 Add 5. S Now Try Exercises 17 and 19. n thTerm of an Arithmetic Sequence In an arithmetic sequence with first term a1 and common difference d, the nth term an is given by the following. an =a1 + 1n −12d
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