A sequence ${{u}_{0}}$, ${{u}_{1}}$ , ${{u}_{2}}$, … is given by ${{u}_{0}}=1$ and ${{u}_{n}}={{u}_{

Question

A sequence ${{u}_{0}}$, ${{u}_{1}}$ , ${{u}_{2}}$, … is given by ${{u}_{0}}=1$ and ${{u}_{n}}={{u}_{n-1}}+{{n}^{2}}+3n$ for $n\ge 1$. By considering $\sum\limits_{r=1}^{n}{\left( {{u}_{r}}-{{u}_{r-1}} \right)}$, find a formula for ${{u}_{n}}$ in terms of $n$. Leave your answer in the form $\frac{n\left( n+1 \right)\left( n+A \right)}{B}+C$, where $A$, $B$ and $C$ are real constants. [4] [You may use the result $\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n}{6}\left( n+1 \right)\left( 2n+1 \right)}$.]

Accepted Answer

$u_n = \displaystyle\frac{n(n+1)(n+5)}{3} + 1$; $A = 5$, $B = 3$, $C = 1$

2022 ASRJC MYCT P1 Q3

2022 ASRJC MYCT P1 Q3

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