The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, \({{u}_

Question

The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is given by ${{S}_{n}}=a\left( {{3}^{n}} \right)+bn+c$, where $a$, $b$ and $c$ are constants and $n\ge 1$. Given that ${{u}_{1}}=5$, ${{u}_{2}}=9$ and ${{u}_{3}}=33$, find $a$, $b$ and $c$ .[4] Show that ${{u}_{n+1}}=A~\left( {{3}^{n}} \right)+B$, where $A$ and $B$ are constants to be determined.[3] Using your answer in part (ii), find $\sum\limits_{r=2}^{n}{{{u}_{r+1}}}$ in terms of $n$. (You need not simplify your answer.)[3]

Accepted Answer

(i) $a=2$, $b=-3$, $c=2$ (ii) $A=4$, $B=-3$ (iii) $6\cdot3^n-3n-15$

2020 EJC P1 Q7

2020 EJC P1 Q7

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