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

L e t N \in Z . S h o w t h a t

\underset{k = 1}{\sum^{N - 1}} \frac{N - k}{s i n (k π / N)} > 2 N - 2 + \underset{k = 2}{\sum^{N - 1}} \frac{N - k}{s i n ((k - 1) π / (N - 1))}

i f, a n d o n l y i f, N ⩾ 12

Commented by prakash jain last updated on 17/Oct/16

T_{N} = \underset{k = 2}{\sum^{N - 1}} \frac{N - k}{s i n ((k - 1) π / (N - 1))}

S_{N} = \underset{k = 1}{\sum^{N - 1}} \frac{N - k}{s i n (k π / N)}

T_{N} = S_{N - 1}

t o p r o v e

S_{N} - S_{N - 1} > 2 N - 2 i f N ⩾ 12

S_{N} - S_{N - 1} < 2 N - 2 i f N < 12

w i l l c o n t i n u e s o o n