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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
\(B=\dfrac{1}{2}+\dfrac{2}{2^2}+\dfrac{3}{2^3}+.......+\dfrac{99}{2^{99}}+\dfrac{100}{2^{100}}\)
\(\Leftrightarrow2B=1+\dfrac{1}{2^2}+\dfrac{2}{2^3}+\dfrac{3}{2^4}+........+\dfrac{98}{2^{99}}+\dfrac{99}{2^{100}}\)
\(\Leftrightarrow2B-B=\left(1+\dfrac{1}{2^2}+\dfrac{2}{2^3}+........+\dfrac{99}{2^{100}}\right)-\left(\dfrac{1}{2}+\dfrac{2}{2^2}+......+\dfrac{100}{2^{100}}\right)\)
\(\Leftrightarrow B=\dfrac{1}{2}+\dfrac{1}{2^2}+..........+\dfrac{1}{2^{100}}-\dfrac{100}{2^{100}}\)
Đặt :
\(A=\dfrac{1}{2}+\dfrac{1}{2^2}+.....+\dfrac{1}{2^{100}}\)
\(\Leftrightarrow2A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+........+\dfrac{1}{2^{99}}\)
\(\Leftrightarrow2A-A=\left(1+\dfrac{1}{2}+......+\dfrac{1}{2^{99}}\right)-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+.....+\dfrac{1}{2^{100}}\right)\)
\(\Leftrightarrow A=1-\dfrac{1}{2^{100}}\)
\(\Leftrightarrow B=1-\dfrac{1}{2^{100}}-\dfrac{100}{2^{100}}\)
\(\Leftrightarrow B=\dfrac{2^{100}-101}{2^{100}}\)
S = 12 + 22 + 32 + ..... + 1002
= 1.(2-1) + 2.(3-1) + .... + 100.(101-1)
= 1.2 + 2.3 + ....... + 100.101 - (1+2+3+...+100)
Đặt A = 1.2 + 2.3 + ... + 100.101
3A = 1.2.(3-0) + 2.3.(4-1) + .... + 100.101.(102-99)
3A = 1.2.3 + 2.3.4-1.2.3 + ... +100.101.102-99.100.101
3A = 100.101.102
A = 100.101.102 : 3 = 33367034
S = 33367034 - 5050
S = 33361984
S = 12 + 22 + 32 + ..... + 1002
= 1.(2-1) + 2.(3-1) + .... + 100.(101-1)
= 1.2 + 2.3 + ....... + 100.101 - (1+2+3+...+100)
Đặt A = 1.2 + 2.3 + ... + 100.101
3A = 1.2.(3-0) + 2.3.(4-1) + .... + 100.101.(102-99)
3A = 1.2.3 + 2.3.4-1.2.3 + ... +100.101.102-99.100.101
3A = 100.101.102
A = 100.101.102 : 3 = 33367034
S = 33367034 - 5050
S = 33361984