Term index in a sequence

Question

Let $t_1, t_2, \dots$ be real numbers such that $t_1 + t_2 + \dots + t_n = 2n^2 + 9n + 13$, for every positive integer $n \ge 2$. If $t_k = 103$, then $k$ equals

Accepted Answer

24. Let $S_n = t_1 + t_2 + \dots + t_n = 2n^2 + 9n + 13$ for $n \ge 2$. For $n \ge 3$, $t_n = S_n - S_{n-1}$. $$t_n = (2n^2 + 9n + 13) - (2(n-1)^2 + 9(n-1) + 13)$$ $$t_n = 2n^2 + 9n + 13 - (2n^2 - 4n + 2 + 9n - 9 + 13)$$ $$t_n = 4n + 7$$ We are given $t_k = 103$: $$4k + 7 = 103$$ $$4k = 96 \implies k = 24$$

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Term index in a sequence

Explanation

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