Question

Tính tổng A theo cách nhanh nhất :

 

A = 5 + \(\frac{5}{1+2}+\frac{5}{1+2+3}+\frac{5}{1+2+3+4}+.....+\frac{5}{1+2+3+...+100}\)

Answer 1

\(A=5+\frac{5}{1+2}+\frac{5}{1+2+3}+...+\frac{5}{1+2+3+...+100}\)
 

Answer 2

A = \(5+\frac{5}{1+2}+\frac{5}{1+2+3}+...+\frac{5}{1+2+3+..+100}\)

\(=5x\left(1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+...+100}\right)\)

\(=5x\left(1+\frac{1}{3}+\frac{1}{6}+...+\frac{1}{5050}\right)\)

\(=2x5x\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{10100}\right)\)

\(=10x\left(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{100x101}\right)\)

\(=10x\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{100}-\frac{1}{101}\right)\)

\(=10x\left(1-\frac{1}{101}\right)\)

\(=10x\frac{100}{101}\)

\(=\frac{1000}{101}\)

Answer 3

Mình cảm ơn 2 bạn nha !!!

Answer 4

\(A=5+\frac{5}{1+2}+\frac{5}{1+2+3}+...+\frac{5}{1+2+3+...+100}\)

\(=5\left(1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+...+100}\right)\)

\(=5\left(1+\frac{1}{3}+\frac{1}{6}+...+\frac{1}{5050}\right)\)

\(=2.5\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{10100}\right)\)

\(=10\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{100.101}\right)\)

\(=10\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{100}-\frac{1}{101}\right)\)

\(=10\left(1-\frac{1}{101}\right)\)

\(=10.\frac{100}{101}\)

\(=\frac{1000}{101}\)