The sum of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is giv

Question

The sum of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is given by ${{S}_{n}}={{\mathrm{e}}^{2}}{{n}^{2}}-3n$. Show that ${{u}_{n}}={{\mathrm{e}}^{2}}\left( 2n-1 \right)-3$.[2] Hence show that the sequence is an arithmetic progression.[2] State the first $3$ odd-numbered terms, ${{u}_{1}}$, ${{u}_{3}}$, and ${{u}_{5}}$. Hence find the least value of $n$ such that the sum of the first $n$ odd-numbered terms is more than $500$.[3] Find the range of values of $x$ for which the following geometric series converges, $\mathrm{e}-{{\mathrm{e}}^{\frac{1}{x}+1}}+{{\mathrm{e}}^{\frac{2}{x}+1}}-{{\mathrm{e}}^{\frac{3}{x}+1}}+...$[2] Given that $x=-1$, find the sum of all the terms of this geometric series after, and including, the ${{8}^{\mathrm{th}}}$ term.[3]

Accepted Answer

(a)(i) $u_n=e^2(2n-1)-3$. (ii) AP with common difference $2e^2$. (iii) $u_1=e^2-3$, $u_3=5e^2-3$, $u_5=9e^2-3$; least $n=7$. (b)(i) $x<0$. (ii) $-\frac1{e^5(e+1)}$ or $-0.00181$ ($3$ s.f.).

2023 ACJC Promo Q8

2023 ACJC Promo Q8

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