Answer
$12$
Work Step by Step
Find the value of $n$ for the sum $\sum\limits_{i =1}^{n}i=78$
$\sum\limits_{i =1}^{n}i=\frac{n(n+1)}{2}$
Thus,
$\frac{n(n+1)}{2}=78$
$n^{2}+n=78\times2$
$n^{2}+n=156$
$n^{2}+n-156=0$
$(n+13)(n-12)=0$
$n=-13, 12$
Through neglecting the negative value of $n$, we have
$n=12$
Hence, $n=12$
