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Let-N-Z-Show-that-k-1-N-1-N-k-sin-kpi-N-gt-2N-2-k-2-N-1-N-k-sin-k-1-pi-N-1-if-and-only-if-N-12-




Question Number 8595 by diofanto last updated on 17/Oct/16
Let N ∈ Z. Show that  Σ_(k=1) ^(N−1) ((N−k)/(sin(kπ/N))) > 2N−2+Σ_(k=2) ^(N−1) ((N−k)/(sin((k−1)π/(N−1))))  if, and only if, N ≥ 12
$${Let}\:{N}\:\in\:\mathbb{Z}.\:{Show}\:{that} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{N}−\mathrm{1}} {\sum}}\frac{{N}−{k}}{{sin}\left({k}\pi/{N}\right)}\:>\:\mathrm{2}{N}−\mathrm{2}+\underset{{k}=\mathrm{2}} {\overset{{N}−\mathrm{1}} {\sum}}\frac{{N}−{k}}{{sin}\left(\left({k}−\mathrm{1}\right)\pi/\left({N}−\mathrm{1}\right)\right)} \\ $$$${if},\:{and}\:{only}\:{if},\:{N}\:\geqslant\:\mathrm{12} \\ $$
Commented by prakash jain last updated on 17/Oct/16
T_N =Σ_(k=2) ^(N−1) ((N−k)/(sin((k−1)π/(N−1))))  S_N =Σ_(k=1) ^(N−1) ((N−k)/(sin(kπ/N)))  T_N =S_(N−1)   to prove  S_N −S_(N−1) >2N−2 if N≥12  S_N −S_(N−1) <2N−2 if N<12  will continue soon
$${T}_{{N}} =\underset{{k}=\mathrm{2}} {\overset{{N}−\mathrm{1}} {\sum}}\frac{{N}−{k}}{{sin}\left(\left({k}−\mathrm{1}\right)\pi/\left({N}−\mathrm{1}\right)\right)} \\ $$$${S}_{{N}} =\underset{{k}=\mathrm{1}} {\overset{{N}−\mathrm{1}} {\sum}}\frac{{N}−{k}}{{sin}\left({k}\pi/{N}\right)} \\ $$$${T}_{{N}} ={S}_{{N}−\mathrm{1}} \\ $$$${to}\:{prove} \\ $$$${S}_{{N}} −{S}_{{N}−\mathrm{1}} >\mathrm{2}{N}−\mathrm{2}\:{if}\:{N}\geqslant\mathrm{12} \\ $$$${S}_{{N}} −{S}_{{N}−\mathrm{1}} <\mathrm{2}{N}−\mathrm{2}\:{if}\:{N}<\mathrm{12} \\ $$$${will}\:{continue}\:{soon} \\ $$

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