How would I go about finding the sum of a non-infinite summation?

Question

I am given: $\sum_{k=1}^n 2^{n(1+k)}$ and I am honestly at a loss on how to proceed. I'm thinking to use a geometric series formula, but the index starts at k=1, and there is a "n" in my summation. How would I shift and proceed?

Thank you

Answer 1

Since $2^{n(1+k)}=2^n2^{nk}$ and $n$ is not the indexing variable, we may rewrite the series as $2^n\sum_{k=1}^{n}2^{nk}$. Next, let $2^n=t$; then the series becomes $2^n\sum_{k=1}^nt^k=t+t^2+t^3+...+t^n$.

From here it is rather straightfoward. Factor out a $t$ from the series, so that you have $t(1+t+t^2+...+t^{n-1})$. By the sum of a geometric series formula, the larger factor is equal to $\frac{1-t^n}{1-t}$. Re-multiply this with the factor of $t$ to get $\frac{t-t^{n+1}}{1-t}$, and then substitute $t$ with $2^n$ to get $\frac{2^n-(2^n)^{n+1}}{1-2^n}=\frac{2^n-2^{n^2+n}}{1-2^n}$.

Finally, re-introduce the $2^n$ that we factored out at the beginning. This leaves you with $\frac{2^{2n}-2^{n^2+2n}}{1-2^n}$.

And that's it!