Tính A=1+3/2^3+4/2^4+5/2^5+...+100/2^100

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

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

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

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

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...

A=1+3/2^3+4/2^4+5/2^5+...100/2^100 
1/2*A = 1/2 + 3/2^4 + 4/2^5 +....+ 99/2^100 + 100/2^101 

A- A/2 = 1/2A =1/2 + 3/2^3 + 1/2^4 +...+1/2^100 - 100/2^101= 

= [1/2+1/2^2 +1/2^3 +...+1/2^100] -100/2^101 (Do 3/2^3 = 1/2^2 +1/2^3) 

=[1-(1/2)^101]/(1-1/2) -100/2^101 = 

=(2^101 -1)/2^100 - 100/2^101 

=> A= (2^101 -1)/2^99 - 100/2^100