Question

Tính tổng S = 1 + \(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\)

~ Các bn ơi help me !! ~

Answer 1

\(\frac{1}{2}S=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{101}}\)

=> \(\frac{1}{2}S-S=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{100}}+\frac{1}{2^{101}}-\frac{1}{2^{100}}-...-\frac{1}{2}-1\)

<=> \(\frac{-1}{2}S=\frac{1}{2^{101}}-1\)

<=> \(S=2-\frac{1}{2^{100}}\)

Answer 2

Ta có : 

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

\(\Rightarrow2S=2+1+\frac{1}{2}+...+\frac{1}{2^{99}}\left(2\right)\)

Lấy (2) - (1) ta được :

\(S=2-\frac{1}{2^{100}}=\frac{2^{101}-1}{2^{100}}\)