alleryHow many terms of the AP $24, 21, 18, \ldots$ must be taken so that their sum is 78?
AOnly 4
BOnly 13
CEither 4 or 13
DNo such number of terms exists
Answer & Solution
Correct answer: C. Either 4 or 13
Use the sum formula $S_n=\frac{n}{2}[2a+(n-1)d]$ with $a=24$ and $d=-3$.
$$78=\frac{n}{2}[48+(n-1)(-3)] = \frac{n}{2}(51-3n).$$
So $156=n(51-3n)$, giving $3n^2-51n+156=0$. Dividing by 3, $n^2-17n+52=0=(n-4)(n-13)$. Hence $n=4$ or $n=13$, so both are possible.
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