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# In an AP: (v) given d equals 5, S9 equals 75, find a and a9 .

Q : 3    In an AP:

(v) given $\small d=5, S_9=75$, find $\small a$ and $\small a_9$.

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It is given that
$\small d=5, S_9=75$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{9} = \frac{9}{2}\left \{ 2\times(a) +(9-1)5\right \}$
$\Rightarrow 75= \frac{9}{2}\left \{ 2a +40\right \}$
$\Rightarrow 150= 18a+360$
$\Rightarrow a = -\frac{210}{18}=-\frac{35}{3}$
Now,
$a_{9} = a+ 8d = -\frac{35}{3} + 8(5)= -\frac{35}{3}+40 = \frac{-35+120}{3}= \frac{85}{3}$

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