How did $-(2^{k-1})-(2^{k-2}) -\dotsb-(2^0)$ become $-2^k+1$?

Question

I have a question, how was the geometric series collapsed to be in the form of $2^{k+1}$?

Answer 1

You mean "How did $(2^{k-1})+(2^{k-2}) \ldots (2^0)$ became $2^{k}-1$?" right?

Well simply use the formula for the sum of geometric terms:$$1+q+q^2+\cdots+ q^n=\dfrac{q^{n+1}-1}{q-1}$$

For $q=2$ and $n=k-1$ u get:$$1+2+2^2+\cdots+2^{k-1}=\dfrac{2^k-1}{2-1}=2^k-1$$

Answer 2

It wasn't collapsed to $2^{k+1}$ nor to $2^k +1$. It was collapsed to $2^k -1$, as of course it should be (I presume you have studied geometric series or you wouldn't have used the term). You may have misread it as $2^k +1$, but don't forget the entire series is negated!

Answer 3

You're missing some minus signs in your question. The identity they used is $$2^{k-1}+2^{k-2}+\dots+2^0 = 2^k-1$$ This follows trivially from the fact that $2^k-1 = 2^{k-1} + (2^{k-1}-1)$.