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A sequence has its first term a₁ = 6 and its r-th term given by a_r = 3*a_(r-1) + 6^r for r = 2, 3,..., n. For some positive integer n, the sum of the first n terms equals (1/5)*(n² - 18n + 84)*(4*6ⁿ - 5*3ⁿ + 1). Find n.
- 9
- 6
- 7
- 8
Correct answer: 9
Solution
Solving a_r = 3*a_(r-1) + 6^r with a₁ = 6: homogeneous solution C*3^r, particular solution D*6^r gives D = 2. So a_r = C*3^r + 2*6^r. Using a₁ = 6: C = -2. Thus a_r = 2*6^r - 2*3^r. Sₙ = 2*(6¹+...+6ⁿ) - 2*(3¹+...+3ⁿ) = (12/5)*(6ⁿ - 1) - (3*(3ⁿ - 1)) =... Matching with the given expression uniquely determines n = 9.
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