An arithmetic sequence ${{a}_{1},{{a}_{2}},{{a}_{3}},...}$ has common difference $d$, where&nbsp

Question

An arithmetic sequence ${{a}_{1},{{a}_{2}},{{a}_{3}},...}$ has common difference $d$, where $d<0$. The sum of the first $n$ terms of the sequence is denoted by ${{S}_{n}}$. Given that $\left| {{a}_{8}} \right|=\left| {{a}_{13}} \right|$, find the value of $n$ for which ${{S}_{n}}$ is maximum.[4] The terms $u_1,u_2$ and $u_3$ are three consecutive terms of a geometric progression. It is given that $u_1,u_2$ and $u_3-32$ form an arithmetic progression, and that $u_1,u_2-4$ and $u_3-32$ form another geometric progression. Find the possible values of $u_1,u_2$ and $u_3$.[6]

Accepted Answer

(a) $10$ (b) ${{u}_{1}}=2,{{u}_{2}}=10$ and ${{u}_{3}}=50$ or ${{u}_{1}}=\frac{2}{9},{{u}_{2}}=\frac{26}{9}$ and ${{u}_{3}}=\frac{338}{9}$

2021 RI Promo Q7

2021 RI Promo Q7

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