SECTION 12.2 Arithmetic Sequences 871 Finding a Recursive Formula for an Arithmetic Sequence The 8th term of an arithmetic sequence is 75, and the 20th term is 39. (a) Find the first term and the common difference. (b) Find a recursive formula for the sequence. (c) What is the n th term of the sequence? Solution EXAMPLE 5 (a) The n th term of an arithmetic sequence is ( ) = + − a a n d 1 . n 1 As a result, { = + = = + = a a d a a d 7 75 19 39 8 1 20 1 This is a system of two linear equations containing two variables, a1 and d, which can be solved by elimination. Subtracting the second equation from the first gives − = = − d d 12 36 3 With d 3, =− use a d7 75 1 + = to find that a d 757 7573 96. 1 ( ) = − = − − = The first term is = a 96, 1 and the common difference is = − d 3. (b) Using formula (1), a recursive formula for this sequence is = = − − a a a 96 3 n n 1 1 (c) Using formula (2), the n th term of the sequence { } an is ( ) ( )( ) = + − = + − − = − a a n d n n 1 96 1 3 99 3 n 1 Finding a Particular Term of an Arithmetic Sequence Find the 41st term of the arithmetic sequence: 2, 6, 10, 14, 18, . . . Solution EXAMPLE 4 The first term of the arithmetic sequence is = a 2, 1 and the common difference is = d 4. By formula (2), the n th term is ( ) = + − = − a n n 2 1 4 4 2 n ( ) = + − a a n d 1 n 1 The 41st term is = ⋅ − = − = a 441 2 164 2 162 41 Now Work PROBLEM 25 Now Work PROBLEMS 17 AND 31 3 Find the Sum of an Arithmetic Sequence The next theorem gives two formulas for finding the sum of the first n terms of an arithmetic sequence. Exploration Graph the recursive formula from Example 5, = a 96, 1 = − − a a 3, n n 1 using a graphing utility. Conclude that the graph of the recursive formula behaves like the graph of a linear function. How is d, the common difference, related to m, the slope of a line? THEOREM Sum of the First n Terms of an Arithmetic Sequence Suppose { } an is an arithmetic sequence with first term a1 and common difference d. The sum Sn of the first n terms of { } an may be found in two ways: [ ] ( ) = + + + + = + − S a a a a n a n d 2 2 1 n n 1 2 3 1 (3) ( ) = + n a a 2 n 1 (4)
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