1. ## Finding the Sum

$\sum ^m _{j=1} 2^j-2^{j-1}$
$[2^1-2^{m+1}][1-2]^{-1}-[2^1-2^m][1-2]^{-1}$
$2^{m+1}-2-2^m+2$
$2^m$

Can someone pin-point where I went wrong?

2. Originally Posted by quantoembryo
$\sum ^m _{j=1} 2^j-2^{j-1}$
$[2^1-2^{m+1}][1-2]^{-1}-[2^1-2^m][1-2]^{-1}$
$2^{m+1}-2-2^m-2$
$2^m$

Can someone pin-point where I went wrong?
$2^j-2^{j-1}=2^{j-1} (2-1)=2^{j-1}$

3. Okay, but why cannot one merely apply the summation formula from the very beggining, like I had done?

4. Originally Posted by quantoembryo
Okay, but why cannot one merely apply the summation formula from the very beggining, like I had done?
$\sum ^m _{j=1} 2^j-2^{j-1}=\sum ^m _{j=1}2^j+\sum ^m _{j=1}2^{j-1}=...$

5. Whoops, just realized I made a technical error. I see where I went wrong. Thanks!

6. Just a side note: this sum telescopes.