Write $3-6+12-...$ using sigma notation in $2$ different ways.

Question

"Write $3-6+12-...$ using sigma notation in $2$ different ways."

I have got the first way, I simply do the infinite sum of $f(k)$ from $k=1$ to infinity, where $f(k) = 3(-2)^{(k-1)}$.

But I don't know how to find the second way. All I am observing is that the common ratio is $-2$, but I have implemented that in my first way. Maybe there is some arithmetic way instead of a geometric way....?

Any help please?

Answer 1

Your second approach is a fine one. The general term is $(-3)4^{k-1}$. Finite sums should agree. The infinite sums using the geometric series formula may not because the series is not convergent. The absolute value of the ratio is greater than $1$ in both cases.