Answer
$n=50$
Work Step by Step
We are told that we have an arithmetic sequence with:
$a=5$ and $d=2$
We know that the sum of an arithmetic sequence is given by:
$S_{n}= \frac{n}{2}[2a+(n-1)d]$
We need this to equal $2700$:
$2700= \frac{n}{2}[10+2(n-1)]$
$5400=10n+2n^{2}-2n$
$10n+2n^{2}-2n-5400=0$
$2n^{2}+8n-5400=0$
$n^{2}+4n-2700=0$
$(n-50)(n+54)=0$
$(n-50)=0$ or $(n+54)=0$
$n=50$ or $n=-54$
But $n$ can not be negative, so the answer is $n=50$.
