Question

Вычислите сумму 1/1*3 + 1/2*4 + 1/3*5 + ... + 1/98*100

Answer 1

1/n*(n + 2) = 1/2*(n + 2 - n)/n(n+2) = 1/2*(1/n - 1/(n + 2))

1/1*3 + 1/2*4 + 1/3*5 + ...+ 1/97*99 + 1/98*100 =

= 1/2*(1 - 1/3 + 1/2 - 1/4 + 1/3 - 1/5 + 1/4 - 1/6 + ..... + 1/95 - 1/97 + 1/96 - 1/98 + 1/97 - 1/99 + 1/98 - 1/100) = 1/2*(1 + 1/2 - 1/99 - 1/100) =  

1/2*(9900 +  4950 - 100 - 99)/9900 = 1/2*14651/9900 = 14651/19800