How to apply property to a combined sum? I know I don't have the vocabulary to describe this problem.  
The easiest way to describe is by showing.
I have the following sum from a recursion:
$$
\sum_{i = 1}^k{a^{k-i}b^i*i}
$$
And the following two properties:
$$
\sum_{i=1}^k{a^{k-i}} = \frac{x^k-1}{x-1}\\
\sum_{i=1}^k{i*b^i} = \frac{b-(k+1)b^{k+1} + k*b^{k+2}}{1-b^2}\\
$$
How would I go about applying the two properties here?  Is it possible?

Hint. One may just write
$$
\sum_{i = 1}^k{a^{k-i}b^i\cdot i}=a^k\sum_{i = 1}^ki\cdot{\left(\frac{b}{a}\right)^i}, \quad a\neq b.
$$
