Question

Tính hợp lý các tổng và tích sau:

1) \(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\)

2) \(\frac{10}{56}+\frac{10}{140}+\frac{10}{260}+...+\frac{10}{1400}\)

3) \(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+...+\frac{1}{3^{300}}\)

Answer 1

\(\frac{10}{56}+\frac{10}{140}+\frac{10}{260}+....+\frac{10}{1400}\)

\(=\frac{5}{4.7}+\frac{5}{7.10}+\frac{5}{10.13}+.....+\frac{5}{25.28}\)

\(=\frac{5}{3}.\left(\frac{3}{4.7}+\frac{3}{7.10}+\frac{3}{10.13}+....+\frac{3}{25.28}\right)\)

\(=\frac{5}{3}.\left(\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+.....+\frac{1}{25}-\frac{1}{28}\right)\)

\(=\frac{5}{3}\left(\frac{1}{4}-\frac{1}{28}\right)\)

\(=\frac{5}{3}.\frac{3}{14}=\frac{5}{14}\)

Answer 2

a)Đặt A=Tổng trên, ta có:

\(2A=2\left(1+\frac{1}{2}+...+\frac{1}{2^{100}}\right)\)

\(2A=2+1+...+\frac{1}{2^{99}}\)

\(2A-A=\left(2+1+...+\frac{1}{2^{99}}\right)-\left(1+\frac{1}{2}+...+\frac{1}{2^{100}}\right)\)

\(A=2-\frac{1}{2^{100}}\)

b)có đứa làm rồi

c)Đặt C=Tổng trên 

\(3C=3\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{300}}\right)\)

\(3C=1+\frac{1}{3}+...+\frac{1}{3^{299}}\)

\(3C-C=\left(1+\frac{1}{3}+...+\frac{1}{3^{299}}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{300}}\right)\)

\(2C=1-\frac{1}{3^{300}}\)

\(C=\frac{1-\frac{1}{3^{300}}}{2}\)