A sequence ${{u}_{0}}$, ${{u}_{1}}$, ${{u}_{2}}$, … is defined by ${{u}_{0}}=10$ and ${{u}_{n}}={{u}

Question

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 ${{u}_{2}}=5$. Show that $A=2$ and $B=-5$. [1] 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.]

Accepted Answer

(a) $A=2$, $B=-5$; (b) $u_n=10+\frac16n(n+1)(4n-13)$, so $C=10,D=4,E=-13$

2023 CJC J2 MYE P1 Q3

2023 CJC J2 MYE P1 Q3

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