A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is defined by ${{u}_{1}}=2$ and \({{u}_{

Question

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is defined by ${{u}_{1}}=2$ and ${{u}_{r+1}}={{u}_{r}}+{{r}^{2}}+r+{{2}^{r}}$, where $r\ge 1$. Given that $\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{1}{6}n\left( n+1 \right)\left( 2n+1 \right)}$, by considering $\sum\limits_{r=1}^{n-1}{\left( {{u}_{r+1}}-{{u}_{r}} \right)}$, find an expression for ${{u}_{n}}$ in terms of $n$.[6] Describe the behaviour of ${{u}_{r}}$.[1] Evaluate $\sum\limits_{r=4}^{20}{{{u}_{r+2}}}$.[1]

Accepted Answer

(a) $u_n = \frac{n(n-1)(n+1)}{3} + 2^n$ (b) As $n$ increases, ${{u}_{n}}$ increases and diverges. (c) $8\,409\,726$

2023 NYJC P1 Q7

2023 NYJC P1 Q7

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