# (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

$$100 \sum_{k=0}^6 \left(-\frac12 \right)^k=67.1875$$
$$100 \sum_{k=0}^6 \left(\frac12 \right)^k=198.4375$$
• You're welcome. That $+25$ for me... – draks ... Dec 15 '13 at 23:22