Question

Calculati (3+5/2+7/3+9/4+...+2019/1009)-(1+1/2+1/3+1/4+...+1/1009)

Answer 1

Răspuns:

2018

Explicație pas cu pas:

termenul general al sirului din prima paranteza este

$\frac{2n+1}{n}=\frac{2n}{n} +\frac{1}{n} =2+ \frac{1}{n} .$

Deci fiecare termen a primei sume se va scrie ca o suma

$(3+\frac{5}{2} +\frac{7}{3}+\frac{9}{4}+...+\frac{2019}{1009}) - (1+\frac{1}{2}+\frac{1}{3} + ... + \frac{1}{1009})=(2+1+2+\frac{1}{2}+2+ \frac{1}{3} +2+\frac{1}{4}+...+2+\frac{1}{1009}) - (1+\frac{1}{2}+\frac{1}{3} + ... + \frac{1}{1009})=2*1009+(1+\frac{1}{2}+\frac{1}{3} + ... + \frac{1}{1009}) - (1+\frac{1}{2}+\frac{1}{3} + ... + \frac{1}{1009})=2018$