2023 CJC J2 MYE P1 Q3
A sequence ${{u}_{0}}$, ${{u}_{1}}$, ${{u}_{2}}$, … is defined by ${{u}_{0}}=10$ and ${{u}_{n}}={{u}_{n-1}}+A{{n}^{2}}+Bn$ where $A$ and $B$ are constants and $n\ge 1$. It is further given that ${{u}_{1}}=7$ and ${{u}_{2}}=5$.
(a)
Show that $A=2$ and $B=-5$.
[1]
(b)
By considering $\sum\limits_{r=1}^{n}{\left( {{u}_{r}}-{{u}_{r-1}} \right)}$, find a formula for ${{u}_{n}}$ in the form $C+\frac{1}{6}n\left( n+1 \right)\left( Dn+E \right)$ where $C$, $D$ and $E$ are constants to be determined.
[5]
[The formula for $\sum\limits_{r=1}^{n}{{{r}^{2}}}=\frac{1}{6}n\left( n+1 \right)\left( 2n+1 \right)$ may be quoted without proof.]
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