Question

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

Answer 1

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

\(\Rightarrow2A=2+\frac{3}{2^2}+\frac{4}{2^3}+...+\frac{100}{2^{99}}\)

\(\Rightarrow2A-A=\left(2+\frac{3}{2^2}+\frac{4}{2^3}+...+\frac{100}{2^{99}}\right)-\left(1+\frac{3}{2^3}+\frac{4}{2^4}+...+\frac{100}{2^{100}}\right)\)

\(\Rightarrow A=\left(2-1\right)+\frac{3}{2^2}+\left(\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{99}}\right)-\frac{100}{2^{100}}\)

Mà \(\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{99}}=\frac{1}{2^2}-\frac{1}{2^{99}}\) ( tự tính )

\(\Rightarrow A=1+\frac{3}{2^2}+\frac{1}{2^2}-\frac{1}{2^{99}}-\frac{100}{2^{100}}\)

\(\Rightarrow A=1+1-\frac{1}{2^{99}}-\frac{100}{2^{100}}\)

\(\Rightarrow A=2-\frac{1}{2^{99}}-\frac{100}{2^{100}}\)

Vậy...

Answer 2

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

=\(\frac{1}{2}+\frac{2}{2^3}+\frac{3}{2^3}+....+\frac{100}{2^{100}}\)+\(\frac{1}{2}\left(\frac{1}{2}+\frac{2}{2^2}+...+\frac{99}{2^{99}}\right)\)

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

<=> \(\frac{1}{2}A=1-\frac{1}{2^{100}}-\frac{100}{2^{100}}=1-\frac{101}{2^{100}}\)

=\(\frac{2^{100}-101}{2^{100}}\Rightarrow A=2\)