Question

Tính nhanh, tính bằng cách hợp lí:

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

Answer 1

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

\(S=3\left(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{5050}\right)\)

\(S=3.\frac{1}{2}\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{10100}\right)\)

\(S=\frac{3}{2}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{100.101}\right)\)

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

\(S=\frac{3}{2}\left(1-\frac{1}{101}\right)\)

\(S=\frac{3}{2}.\frac{100}{101}=\frac{150}{101}\)

Answer 2

S = 3 + 1 + 1/2 +....