Формулы для нахождения сумм

содержание справочника

1. $1+2+3+...+n=\displaystyle\frac{n(n+1)}{2}$
2. $p+(p+1)+(p+2)+...+(p+n)=\displaystyle\frac{(n+1)(2p+n)}{n}$
3. $1^2+2^2+3^2+...+n^2=\displaystyle\frac{n(n+1)(2n+1)}{6}$
4. $1^3+2^3+3^3+...+n^3=\displaystyle\frac{n^2(n+1)^2}{4}$
5. $1^4+2^4+3^4+...+n^4=\displaystyle\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$
6. $1+3+5+...+(2n-1)=n^2$
7. $1^2+3^2+5^2+...+(2n-1)^2=\displaystyle\frac{n(4n^2-1)}{3}$
8. $1^3+3^3+5^3+...+(2n-1)^3=n^2(2n^2-1)$
9. $1\cdot 2+2\cdot 3+...+n(n+1)=\displaystyle\frac{n(n+1)(n+2)}{3}$
10. $1\cdot 4+2\cdot 7+...+n(3n+1)=n(n+1)^2$
11. $1\cdot 2\cdot 3+2\cdot 3\cdot 4+...+n(n+1)(n+2)=\displaystyle\frac{1}{4}n(n+1)(n+2)(n+3)$
12. $1^2-2^2+3^2-4^2+...+(-1)^{n-1}n^2=(-1)^{n-1}\displaystyle\frac{n(n+1)}{2}$
13. $\displaystyle\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+...+\frac{1}{n(n+1)}=\displaystyle\frac{n}{n+1}$
14. $\displaystyle\frac{1}{1\cdot 3}+\frac{1}{3\cdot 5}+...+\frac{1}{(2n-1)(2n+1)}=\displaystyle\frac{n}{2n+1}$
15. $\sin x+\sin 2x+\sin 3x+...+\sin nx=\displaystyle\frac{\sin\frac{n+1}{2}x\cdot\sin\frac{nx}{2}}{\sin\frac{x}{2}}$
16. $\cos x+\cos 2x+\cos 3x+...+\cos nx=\displaystyle\frac{\cos\frac{n+1}{2}x\cdot\sin\frac{nx}{2}}{\sin\frac{x}{2}}$