N2015 P2 Q4

Junior College 2

14 marks

Prove by the method of mathematical induction that \[1\times3\times6+2\times4\times7+3\times5\times8+\cdots+n(n+2)(n+5)=\frac1{12}n(n+1)(3n^2+31n+74).\][6]
1. Show that $\dfrac2{4r^2+8r+3}$ can be expressed as $\dfrac A{2r+1}+\dfrac B{2r+3}$, where $A$ and $B$ are constants to be determined.[1]
2. The sum $\displaystyle\sum_{r=1}^n\frac2{4r^2+8r+3}$ is denoted by $S_n$. Find an expression for $S_n$ in terms of $n$.[4]
3. Find the smallest value of $n$ for which $S_n$ is within $10^{-3}$ of the sum to infinity.[3]

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Answer:(b)(i) $A=1,B=-1$ (ii) $S_n=\frac13-\frac1{2n+3}$ (iii) $499$.

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