Sum of the Arithmetico-Geometric Progression (AGP)

Question

1 + 2•3 + 3•9 + ^… + (k+1)•3^k = ? 

(k is a non-negative integer)

Answer 1

Let S=1+2.3+3.9+....+(k+1).3^k

Multiply by 3

3S=1.3+2.9+....+k3^k+(k+1).3^k+1

Subtract both equations

-2S=(3+3²+3³+...+3^k)-(k+1).3^k+1+1

-2S=[3.(3^k-1)/(3-1)]-(k+1).3^k+1+1

So S=(1/2).{(K+1).3^k+1-([3.(3^k-1)]/2)-1}

Answer 2

k/2*3^(k+1)+1/2