I've reached the correct answer but I don't see how the sum of the first 5 terms in the following geometric sequence is greater than the sum to infinity:
$$a= 9, \, r= \frac{-1}{3}$$
Given: Sum to infinity = 6.75
$S_5=\frac{(9(1-(-1/5)^5))}{4/3} = 6.78$, using $S_n=\frac{a(1-r^n)}{1-r}$
Could it be simply because the sum to infinity will continue to increase marginally?
Thanks in advance,
Matt.
When the terms are positive the sum up to $n$ terms keeps increasing with $n$ so the sum of first five terms has to be less than the sum to $\infty$. But when you have positive and negative terms there is absolutely no reason why sum of first five terms has to be less than the sum to $\infty$.