Let t 1 , t 2 ,… be real numbers such that t 1 + t 2 + … + t n = 2n 2 + 9n + 13, for every positive integer n ≥ 2. If t k = 103, then k equals

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Let t 1 , t 2 ,&hellip; be real numbers such that t 1 + t 2 + &hellip; + t n = 2n 2 + 9n + 13, for every positive integer n &ge; 2. If t k = 103, then k equals

Accepted Answer

24 — Easy The given expression is the partial sum $S_n=t_1+\dots+t_n$. Recover a single term via $t_k=S_k-S_{k-1}$, then set it equal to $103$ and solve for $k$. 1 Express the general term. With $S_n = 2n^2+9n+13$, compute $t_k=S_k-S_{k-1}$. $$\begin{aligned} &t_k = \bigl(2k^2+9k+13\bigr) - \bigl(2(k-1)^2 + 9(k-1) + 13\bigr)\ &\Rightarrow\ = 2\bigl(k^2-(k-1)^2\bigr) + 9\bigl(k-(k-1)\bigr)\ &\Rightarrow\ = 2(2k-1) + 9\ &\Rightarrow\ t_k = 4k + 7 \end{aligned}$$ 2 Solve $t_k=103$. Use the formula fr

CAT 2018 Slot 2 — QA Question 30

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