Given that $\sum\limits_{k=1}^{m}{\left( {{2}^{k}}-200k \right)}0$, show that \({{2}

Question

Given that $\sum\limits_{k=1}^{m}{\left( {{2}^{k}}-200k \right)}>0$, show that ${{2}^{m}}-50{{m}^{2}}-50m-1>0$. Hence find the last value of $m$.[4] Prove that $\frac{1}{\left( r-2 \right)!}-\frac{1}{r!}=\frac{{{r}^{2}}-r-1}{r!}$.[1] Show that $\sum\limits_{r=3}^{n}{\frac{{{r}^{2}}-r-1}{r!}}=\frac{3}{2}-\frac{n+1}{n!}$.[3] Using the result in part (ii), find $\sum\limits_{r=6}^{10}{\frac{{{r}^{2}}-r-1}{r!}}$in the form $\frac{k}{10!}$ where $k$ is an integer to be determined.[3]

Accepted Answer

$m = 14$

2016 TJC Promo Q10

2016 TJC Promo Q10

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