Earlier I was looking to find a closed form expression of the sum: $$\sum_{h=1}^{k-1}\left({2^{h}}{3^{k-h-1}}\right)$$ Wolfram Alpha tells me that this is equivalent to: $$2\times{3^{k-1}}-2^k$$ However it does not explain the method of how it got there, and I cannot seem to find any resources online explaining how expressions can be found for more general cases (although I'm sure they're out there). I was wondering if anyone could give a method for deriving this or point to resources that explain how different sums can be expressed in closed form.
Thanks :)