数列sn=2*2^2+3*2^3+4*2^4+5*2^5.(n+1)*2^n+1怎么化简

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数列sn=2*2^2+3*2^3+4*2^4+5*2^5.(n+1)*2^n+1怎么化简

数列sn=2*2^2+3*2^3+4*2^4+5*2^5.(n+1)*2^n+1怎么化简
数列sn=2*2^2+3*2^3+4*2^4+5*2^5.(n+1)*2^n+1怎么化简

数列sn=2*2^2+3*2^3+4*2^4+5*2^5.(n+1)*2^n+1怎么化简

sn=2*2^2+3*2^3+4*2^4+5*2^5+………+(n+1)*2^n+1

2sn=     2*2^3+3*2^4+4*2^5+5*2^6+………+n*2^(n+1)+(n+1)*2^(n+2)

相减得

sn-2sn=2*2^2+2^3+2^4+2^5+………+2^(n+1)-(n+1)*2^(n+2)

-sn=2^2+2^2+2^3+2^4+2^5+………+2^(n+1)-(n+1)*2^(n+2)

=[2^2+2^3+2^4+2^5+………+2^(n+1)]+2^2-(n+1)*2^(n+2)

=4×(1-2ⁿ)/(1-2)+4-(n+1)*2^(n+2)

=4(2ⁿ-1)+4-(n+1)×2^(n+2)

=4×2ⁿ-4+4-(n+1)×2^(n+2)

=2^(n+2)-(n+1)×2^(n+2)

=2^(n+2)-n×2^(n+2)-2^(n+2)

=-n×2^(n+2)

所以sn=n×2^(n+2)


利用的是错位相减法

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