Question

Hãy viết dạng tổng quát của \(\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)...\left(1-\frac{1}{n}\right)\)

Answer 1

Mk sửa đề chỗ thừa số cuối nhé, có lẽ bn chép sai đề

\(\left(1-\frac{1}{4}\right).\left(1-\frac{1}{9}\right)...\left(1-\frac{1}{n^2}\right)\)

\(=\frac{3}{4}.\frac{8}{9}...\frac{n^2-1}{n^2}\)

\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}...\frac{\left(n-1\right).\left(n+1\right)}{n.n}\)

\(=\frac{1.2...\left(n-1\right)}{2.3...n}.\frac{3.4...\left(n+1\right)}{2.3...n}\)

\(=\frac{1}{n}.\frac{n+1}{2}=\frac{n+1}{2n}\)