Question

How many terms of the A.P: $24,21,18,....$ must be taken so that their sum is $78$?

Answer 1

$\Rightarrow S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

$\Rightarrow$$78=\frac{n}{2}(2a+(n-1\left)d\right)$

$\Rightarrow$$78=\frac{n}{2}\left(2\right(24)+(n+1\left)\right(-3\left)\right)$

$\Rightarrow$$156=n(48-3n+3)$

$\Rightarrow$$156=n(51-3n)$

$\Rightarrow$$156=51n-3n^2$

$\Rightarrow$$3n^2-51n+156=0$

$\Rightarrow$$3n^2-12n-39n+156=0$

$\Rightarrow$$3n(n-4)-3q(n-4)=0$

$\Rightarrow$$n=4$ or $n=\frac{39}{3}=13$