Section I.1.7

The principle of mathematical induction

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Problem I.1.7.1

Let us set

s(n)=sin⁡φ+sin⁡2φ+…+sin⁡nφ,c(n)=cos⁡φ+cos⁡2φ+…+cos⁡nφ. \begin{aligned} & s(n)=\sin \varphi +\sin 2 \varphi +\ldots +\sin n \varphi , \\ & c(n)=\cos \varphi +\cos 2 \varphi +\ldots +\cos n \varphi . \end{aligned}

By induction on nn, prove the formulas

s(n)=sin⁡(nφ/2)sin⁡((n+1)φ/2)sin⁡(φ/2),c(n)=sin⁡(nφ/2)cos⁡((n+1)φ/2)sin⁡(φ/2) s(n)=\frac{\sin (n \varphi / 2) \sin ((n+1) \varphi / 2)}{\sin (\varphi / 2)}, \quad c(n)=\frac{\sin (n \varphi / 2) \cos ((n+1) \varphi / 2)}{\sin (\varphi / 2)}
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Problem I.1.7.2

The following formulas hold:

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(a)

∑k=1nctg⁡2(kπ2n+1)=n(2n−1)3\sum_{k=1}^{n} \operatorname {ctg}^{2}\left(\frac{k \pi }{2 n+1}\right)=\frac{n(2 n-1)}{3};

(b)

∑k=0n(2kk)(2n−2kn−k)=4n\sum_{k=0}^{n}\left(\begin{array}{c}2 k \\ k\end{array}\right)\left(\begin{array}{c}2 n-2 k \\ n-k\end{array}\right)=4^{n}.

Verify their validity at least for n⩽5n \leqslant 5.