Question

求1×2+4×2^2+7×2^3+.+(3n-2)×2^n

Answer 1

设上式为T

则2T=1×2^2+4×2^3+7×2^4+.+(3n-2)×2^(n+1)

-T=T-2T

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

=2-(3n-2)×2^(n+1)+12(1+2+……+2^(n-2))

=2-(3n-2)×2^(n+1)+6(2^n-2)

=(5-3n)2^(n+1)-10

所以T=(3n-5)2^(n+1)+10