Question

Дорешайте 12 умоляю ! Распишите подробнее. 12. начинается 5/1*2 + 13/2*5+25/3*4...

Answer 1

$frac{5}{1cdot 2} +frac{13}{2cdot 3}+ frac{25}{3cdot 4}+...+frac{2n^2+2n+1}{n(n+1)}=\\star ; ; frac{2n^2+2n+1}{n(n+1)}= frac{2n(n+1)+1}{n(n+1)}= frac{2n(n+1)}{n(n+1)}+frac{1}{n(n+1)}=2+frac{1}{n(n+1)}; ; star \\ =underbrace {(2+frac{1}{1cdot 2})}_{1}+underbrace {( 2+frac{1}{2cdot 3})}_{2}+underbrace {(2+frac{1}{3cdot 4})}_{3}+...+underbrace {(2+frac{1}{n(n+1)})}_{n}=\\=2cdot n+Big (frac{1}{1cdot 2}+frac{1}{2cdot 3}+frac{1}{3cdot 4}+...+frac{1}{n(n+1)}Big )=$

$star ; ; frac{1}{n(n+1)} = frac{1}{n} -frac{1}{n+1} ; ; star \\=2cdot n+( frac{1}{1}-frac{1}{2}+frac{1}{2}-frac{1}{3}+frac{1}{3}-frac{1}{4}+...+frac{1}{n-1}-frac{1}{n}+frac{1}{n}-frac{1}{n+1} )=\\=2n+1-frac{1}{n+1}= frac{(2n+1)(n+1)-1}{n+1} = frac{2n^2+3n}{n+1} = frac{n(2n+3)}{n+1}$