# Thread: Evaluation a non-linear series?

1. ## Evaluation a non-linear series?

How do I evaluate $\displaystyle \sum_{n=1}^{20} (2^n - 1)$ without havng to add up each element?

Thanks

2. Originally Posted by chancey
How do I evaluate $\displaystyle \sum_{n=1}^{20} (2^n - 1)$ without havng to add up each element?

Thanks
This is a geometric series.
---
$\displaystyle (2^1+2^2+...+2^{20})-(1+1...+1)$
Thus,
$\displaystyle 2(1+2+2^2+...+2^{19})-20$
Use geometric sum formula,
$\displaystyle 2(2^{20}-1)-20$
Simplify,
$\displaystyle 2^{21}-22$

3. Originally Posted by ThePerfectHacker
This is a geometric series.
---
$\displaystyle (2^1+2^2+...+2^{20})-(1+1...+1)$
Thus,
$\displaystyle 2(1+2+2^2+...+2^{19})-20$
Use geometric sum formula,
$\displaystyle 2(2^{20}-1)-20$
Simplify,
$\displaystyle 2^{21}-22$
Right, but thats not the answer in the back of the book...

I did it this way:
$\displaystyle \sum_{n=1}^{20} (2^n - 1) = \sum_{n=1}^{20} (2^n) - 20$

$\displaystyle \sum_{n=1}^{20} (2^n) = \frac{1-2^{21}}{1-2} - 1 = 2097150$

$\displaystyle \because \sum_{n=0}^{k} (ar^n) = \frac{a(1-r^{k+1})}{1-r}$

$\displaystyle \therefore \sum_{n=1}^{20} (2^n) - 20 = 2097150 - 20 = 2097130$

Which is the same answer as yours, however the answer in the book says $\displaystyle 2097170$. They must of added the 20 instead of subtracted it. OK thanks

4. Originally Posted by chancey
Right, but thats not the answer in the back of the book...

I did it this way:
$\displaystyle \sum_{n=1}^{20} (2^n - 1) = \sum_{n=1}^{20} (2^n) - 20$

$\displaystyle \sum_{n=1}^{20} (2^n) = \frac{1-2^{21}}{1-2} - 1 = 2097150$

$\displaystyle \because \sum_{n=0}^{k} (ar^n) = \frac{a(1-r^{k+1})}{1-r}$

$\displaystyle \therefore \sum_{n=1}^{20} (2^n) - 20 = 2097150 - 20 = 2097130$

Which is the same answer as yours, however the answer in the book says $\displaystyle 2097170$. They must of added the 20 instead of subtracted it. OK thanks
Then the book is inncorect.