For each integer, $\displaystyle r \geq 0 $ , the Mersenne number $\displaystyle M_{r} $ is defined to be $\displaystyle 2^{r} -1 $

Prove by induction that $\displaystyle \sum_{ r = 0}^n M_{r} = M_{n+1} -n -1 $

So I proved it works for 1 first,

I got $\displaystyle 2^{1} -1 = 2^{1+1} -1-1-1 $

so it is true for n =1

assume it s also true for n =k

so $\displaystyle \sum_{r=0}^k 2^{r} -1 = 2^{k+1}-1-k-1 $

and so with n=k+1

$\displaystyle \sum_{r=0}^{k+1} 2^{r}-1 = 1 + 3+ 5....2^{k}-1 + 2^{k+1 +1} -1 $

=

not sure what to do from here...

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