Question

How many terms of the arithmetic series $24+21+18+15+....$, be taken continuously so that their sum is $-351$

Answer 1

In the given arithmetic series, $a=24,d=-3$
Let us find $n$ such that $S_n=-351$
Now, $S_n=\frac{n}{2}[2a+(n-1\left)d\right]=-351$
That is, $\frac{n}{2}\left[2\right(24)+(n-1\left)\right(-3\left)\right]=-351$
$\Rightarrow \frac{n}{2}[48-3n+3]=-351$
$\Rightarrow n(51-3n)=-702$
$\Rightarrow n^2-17n-234=0$
$\Rightarrow (n-26)(n+9)=0$
$\Rightarrow n=26$ or $n=-9$
Here $n$, being the number of terms needed, cannot be negative.
Thus, $26$ terms are needed to get the sum $-351$.