2020 EJC P1 Q7
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 $n\ge 1$.
(i)
Given that ${{u}_{1}}=5$, ${{u}_{2}}=9$ and ${{u}_{3}}=33$, find $a$, $b$ and $c$ .
[4]
(ii)
Show that ${{u}_{n+1}}=A~\left( {{3}^{n}} \right)+B$, where $A$ and $B$ are constants to be determined.
[3]
(iii)
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]
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