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 May 2nd, 2014, 04:38 AM #1 Senior Member   Joined: Nov 2013 Posts: 434 Thanks: 8 sum find sum 2.1+3.2+4.4+5.8+......................18.2^16
 May 2nd, 2014, 12:35 PM #2 Global Moderator   Joined: May 2007 Posts: 6,806 Thanks: 716 Arithmetico-geometric sequence - Wikipedia, the free encyclopedia  Above includes a general formula. Your question is a special case.
May 2nd, 2014, 01:23 PM   #3
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

Hello, mared!

Quote:
 $\displaystyle \text{Find the sum: }\;2\!\cdot\!1+3\!\cdot\!2+4\!\cdot\!4+5\!\cdot\!8 + \cdots + 18\!\cdot\!2^{16}$

We are given: $\displaystyle \;\;\;S \;=\;2\!\cdot\!2^0+3\!\cdot\!2^1+4\!\cdot\!2^2+5\! \cdot\!2^3 + \cdots + 18\!\cdot\!2^{16}$
Multiply by 2: $\displaystyle \;\;2S \;=\; \qquad \quad 2\!\cdot\!2^1 + 3\!\cdot\!2^2 + 4\!\cdot\!2^3 + \cdots + 17\!\cdot\!2^{16} + 18\!\cdot\!2^{17}$

Subtract: $\displaystyle \quad\;\;\; -S \;=\;2 + \underbrace{(2 + 2^2 + 2^3 + \cdots + 2^{16})}_{\text{geo. series}} - 18\!\cdot\!2^{17}$

$\displaystyle \qquad$The geometric series has sum: $\displaystyle \;2(2^{16}-1)$

We have: $\displaystyle \;-S \;=\;2 + 2(2^{16}-1) - 18\!\cdot\!2^{17}$

$\displaystyle \qquad\qquad\; -S \;=\;2 + 2^{17} - 2 - 18\!\cdot\!2^{17}$

$\displaystyle \qquad\qquad\; -S \;=\;-17\!\cdot\!2^{17}$

Therefore: $\displaystyle \;S \;=\;17\!\cdot\!2^{17}$

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