Finding the Sum

**quantoembryo** · January 12th 2011, 10:52 AM

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

Can someone pin-point where I went wrong?

**Also sprach Zarathustra** · January 12th 2011, 11:02 AM

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?

$2^j-2^{j-1}=2^{j-1} (2-1)=2^{j-1}$

**quantoembryo** · January 12th 2011, 11:07 AM

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

**Also sprach Zarathustra** · January 12th 2011, 11:12 AM

Originally Posted by quantoembryo

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

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

**quantoembryo** · January 12th 2011, 11:12 AM

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

**Ackbeet** · January 12th 2011, 11:14 AM

Just a side note: this sum telescopes.

Thread: Finding the Sum