finite summation formula

**jut** · September 19th 2009, 05:11 PM

$\sum_{k=1}^n a^k b^k=?$

**Soroban** · September 19th 2009, 05:46 PM

Hello, jut!

Simplify: . $\sum_{k=1}^n a^k b^k$

We have: . $S \;=\;ab + a^2b^2 + a^3b^3 + \hdots + a^nb^n$

This is a geometric series with: . $\begin{Bmatrix}\text{first term: }ab \\ \text{common ratio: }ab \\ n\text{ terms} \end{Bmatrix}$

Its sum is: . $S \;=\;ab\,\frac{(ab)^n - 1}{ab-1}$

**jut** · September 19th 2009, 05:50 PM

Wow, nice Soroban!

**jut** · September 19th 2009, 10:36 PM

Originally Posted by Soroban

Hello, jut!

Its sum is: . $S \;=\;ab\,\frac{(ab)^n - 1}{ab-1}$

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One more question:

if a=e, and b=-2, I could plug that into the above result and it would be correct?

**bkarpuz** · September 19th 2009, 11:41 PM

Originally Posted by jut

One more question:

if $a=\mathrm{e}$ , and $b=-2$ , I could plug that into the above result and it would be correct?

sure, as long as $ab\neq1$ , because it is a finite sum!
In the case $ab=1$ , you get $\sum_{k=1}^{n}ab=n$ .

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