jmc

Question

Let S_n denote the sum of the first n terms of an arithmetic sequence with common difference 3. If S_3nS_n is a constant that does not depend on n, for all positive integers n, then find the first term.

Accepted Answer

Let a be the first term. Then S_n = n [2a + (n - 1) 3]2and S_3n = 3n [2a + (3n - 1) 3]2,so S_3nS_n = 3n [2a + (3n - 1) 3]2n [2a + (n - 1) 3]2 = 3(2a + 9n - 3)2a + 3n - 3 = 6a + 27n - 92a + 3n - 3.Let this constant be c, so 6a + 27n - 92a + 3n - 3 = c.Then 6a + 27n - 9 = 2ac + 3cn - 3c. Since both sides are equal for all n, the coefficients of n must be equal. In other words, 27 = 3c, so c = 9. then 6a - 9 = 18a - 27. Solving, we find a = 32.