In an AP: d = 5, S_9 = 75, find a and a_9 . - Vidyadip

Question

In an AP: $d = 5, S_9 = 75$, find a and $a_9 $.

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Answer

Here, $d = 5$
$S_9 = 75$
We know that
${S_n} = \frac{n}{2}\left[ {2a + (n - 1)d} \right]$
$ \Rightarrow {S_9} = \frac{9}{2}\left[ {2a + (9 - 1)d} \right]$
$ \Rightarrow {S_9} = \frac{9}{2}\left[ {2a + 8d} \right]$
$ \Rightarrow {S_9} = 9\left[ {a + 4d} \right]$
$ \Rightarrow {S_9} = 9\left[ {a + 4 \times 5} \right]$
$ \Rightarrow S_9 = 9[a + 20]$
$ \Rightarrow 75 = 9a + 180$
$ \Rightarrow 9a = 75 - 180$
$ \Rightarrow 9a = -105$
$ \Rightarrow a = - \frac{{105}}{9}$
$ \Rightarrow a = - \frac{{35}}{3}$
Again, we know that
$a_n = a + (n - 1)d$
$ \Rightarrow $ $a_9 = a + (9 - 1)d$
$ \Rightarrow $ $a_9 = a + 8d$
$ \Rightarrow {a_9} = - \frac{{35}}{3} + 8(5)$
$ \Rightarrow {a_9} = - \frac{{35}}{3} + 40$
$ \Rightarrow {a_9} = \frac{{ - 35 + 120}}{3}$
$ \Rightarrow {a_9} = \frac{{85}}{3}$

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