Question

Sn=1*4+3*4^2+5*4^3+7*4^4+...+(2n-1)*4^n

Answer 1

Sn=1×4+3×4²+5×4³+...+(2n-1)×4ⁿ

4Sn=1×4²+3×4³+...+(2n-3)×4ⁿ+(2n-1)×4^(n+1)

Sn-4Sn=-3Sn=4+2×4²+2×4³+...+2×4ⁿ-(2n-1)×4^(n+1)

=2×(1+4+4²+4³+...+4ⁿ) -(2n-1)×4^(n+1) -6

=2×[4^(n+1)-1]/(4-1) -(2n-1)×4^(n+1) -6

=[(5-6n)×4^(n+1)-20]/3

Sn=[(6n-5)×4^(n+1)+20]/9