A sequence of numbers ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … has a sum ${{S}_{n}}$ where ${{S}_{n}

Question

A sequence of numbers ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … has a sum ${{S}_{n}}$ where ${{S}_{n}}=\sum\limits_{r=1}^{n}{{{u}_{r}}}$. It is given that ${{S}_{n}}=A{{n}^{2}}\left( n+1 \right)-Bn$, where $A$ and $B$ are non-zero constants. Find a simplified expression for ${{u}_{n}}$ in terms of $A$, $B$ and $n$. [2] It is also given that the first and second term of the sequence are $1$ and $9$ respectively. Show that $A=1$ and $B=1$. [3] Hence, find $\sum\limits_{r=1}^{n}{\left[ 3{{\left( r-1 \right)}^{2}}-r \right]}$, giving your answer in the form $a{{n}^{3}}-b{{n}^{2}}$, where $a$ and $b$ are constants to be determined. [3]

Accepted Answer

(i) $u_n=3An^2-An-B$ (ii) $A=1,\ B=1$ (iii) $n^3-2n^2$, so $a=1,\ b=2$

2020 NYJC P1 Q7

2020 NYJC P1 Q7

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