# (Geometric) Sum of $100-50-25-(25/2)+\ldots+ (25/16)$

Determine the sum for this geometric series:

$$100-50+25-(25/2)+\ldots+ (25/16)$$

I found $$7$$ to be the number of terms in this series, and the sum of the series to be $$67.1875.$$, but, the answer book says that the sum of the series is $$198.4375.$$ So far this the work I've done:

$$\dfrac{25}{16}=100\left(-\dfrac{1}{2}\right)^{n-1}$$

$$\implies \left(-\frac{1}{2}\right)^6=\left(-\frac{1}{2}\right)^{n-1}$$

$$\implies n=7$$

And then I used the formula for geometric series, $$s_n=\dfrac{a(r^n-1)}{r-1}$$, but I got $$67.1875$$. Where am I going wrong?

• Do you mean $+25$ instead? – draks ... Dec 15 '13 at 23:10
• yes. sorry typo – Rose Dec 15 '13 at 23:10
• The sum as it is typed now is not in the $198$ range. Either you have not typed the correct expression to be summed, or the answer sheet has a mistake. – André Nicolas Dec 15 '13 at 23:12
• Both the question and the answer are typed properly right now without typos. This is why i asked the question to make sure if the answer sheet was correct or not. Thanks. – Rose Dec 15 '13 at 23:13

## 2 Answers

Maybe you mixed up questions and answers in your book:

$$100 \sum_{k=0}^6 \left(-\frac12 \right)^k=67.1875$$

$$100 \sum_{k=0}^6 \left(\frac12 \right)^k=198.4375$$

• Thanks. I guess the answer book had a typo. – Rose Dec 15 '13 at 23:21
• You're welcome. That $+25$ for me... – draks ... Dec 15 '13 at 23:22

The answer in the book seems to imply that every term is added, rather than alternately added and subtracted. Your answer is closer to what the actual answer would be, if you indeed typed the question correctly.