Solving this summation: $\sum_{i=1}^k i\cdot2^i$ $$\sum_{i=1}^k i\cdot2^i$$
I'm working on a recurrence relation question and now I'm stuck at this point. I have no idea how to simplify this down to something I can work with.  Can I seperate the terms into
$$\sum_{i=1}^k i \cdot \sum_{i=1}^k 2^i$$ and then just use the geometric series? 

No, you can't rewrite
$$\sum_{i=1}^ki\cdot2^i$$
as
$$\sum_{i=1}^ki\cdot\sum_{i=1}^k2^i.$$
In less time that it took to post the question, you could have checked the case $k=2$ on a calculator: is
$$1\cdot2+2\cdot4=(1+2)(2+4)?$$
However, you can write
$$\sum_{i=1}^ki\cdot2^i=\sum_{i=1}^k2^i+\sum_{i=2}^k2^i+\sum_{i=3}^k2^i+\cdots+\sum_{i=k}^k2^i.$$
Right? Now can you take it from there?
