Question

Sn=1+3×2+5×2^2+7×2^3…………+(2n-1)×2^n-1

求Sn

Answer 1

Sn=1×2^0+3×2+5×2^2+7×2^3…………+(2n-1)×2^n-1
2Sn=1×2^1+3×2^2+5×2^3+7×2^4…………+(2n-3)×2^n-1+(2n-1)×2^n
Sn-2Sn=2^0+2*2^1+2*2^2+2*2^3+........+2*2^n-1-(2n-1)×2^n
-Sn=2^0+2*(2^1+2^2+2^3+........+2^n-1)-(2n-1)×2^n
-Sn=1+2*2*[1-2^(n-1)]/(1-2)]-(2n-1)×2^n
-Sn=1+4*[2^(n-1)-1]-(2n-1)×2^n
-Sn=1+2*2^n-4-(2n-1)×2^n
-Sn=2*2^n-3-(2n-1)×2^n
Sn=(2n-1)×2^n-2*2^n+3
Sn=(2n-3)×2^n+3

Answer 2

Sn=1×2^0+3×2+5×2^2+7×2^3…………+(2n-1)×2^n-1

2Sn=1×2^1+3×2^2+5×2^3+7×2^4…………+(2n-3)×2^n-1+(2n-1)×2^n

Sn-2Sn=2^0+2*2^1+2*2^2+2*2^3+........+2*2^n-1-(2n-1)×2^n

-Sn=2^0+2*(2^1+2^2+2^3+........+2^n-1)-(2n-1)×2^n

-Sn=1+2*2*[1-2^(n-1)]/(1-2)]-(2n-1)×2^n

-Sn=1+4*[2^(n-1)-1]-(2n-1)×2^n

-Sn=1+2*2^n-4-(2n-1)×2^n

-Sn=2*2^n-3-(2n-1)×2^n

Sn=(2n-1)×2^n-2*2^n+3

Sn=(2n-3)×2^n+3

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