Sum of n*constant^n from 1 to N

Question

What is the value of the sum for n=1 to n=N of n * constant ^ n?

I have considered Geometric but with the multiplication by n I cannot get this to work.

Answer 1

Hint:

This is an Arithmetico-Geometric Series.

$$\begin{aligned} S_n & = \sum_{n = 1}^N t_k = \sum_{n = 1}^N \left[a + (n - 1) d\right] br^{n - 1} \\ & = ab + [a + d] br + [a + 2 d] br^2 + \cdots + [a + (N - 1) d] br^{N - 1} \\ & = A_1G_1 + A_2G_2 + A_3G_3 + \cdots + A_NG_N, \end{aligned}$$

Here $\begin{bmatrix}a & d \\ b & r\end{bmatrix}=\begin{bmatrix}1 & 1 \\ 1& k\end{bmatrix}$ where $k$ is a constant.

$$\begin{align} S_n & = \frac{ab - (a+nd)\,br^n}{1 - r}+\frac{dbr\,(1 - r^n)}{(1-r)^2} \end{align}$$

Answer 2

We have that

$$\sum_{n=1}^N{nx^n}=x\sum_{n=1}^N{nx^{n-1}}=x\sum_{n=1}^N{\frac{d}{dx}x^n}=x\cdot\frac{d}{dx}\left[\sum_{n=1}^N{x^{n}}\right]$$

There is your geometric series. Multiply your result of the series by $x$ and then differentiate with respect to $x$.