In an A.P. (with usual notations) : given d=5, S_9=75, find a and… - Vidyadip

Question

In an A.P. (with usual notations) : given $d=5, S_9=75$, find a and $a _9$

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Answer

$d =5, S _9=75$
$a _{ n }= a +( n -1) d$
$a _9= a +(9-1) \times 5$
$= a +40...(i)$
$S _9=\frac{n}{2}[2 a+(n-1) d]$
$75=\frac{9}{2}[2 a+8 \times 5]$
$\frac{150}{9}=2 a +40$
$2 a =\frac{150}{9}-40$
$=\frac{50}{3}-40$
$2 a =\frac{-70}{3}$
$\Rightarrow a =\frac{-70}{2 \times 3}$
$a =\frac{-35}{3}$
From (i),
$a g=a+40$
$=\frac{-35}{3}+40$
$=\frac{-35+120}{3}$
$=\frac{85}{3}$
$\therefore a=\frac{-35}{3}, a_9=\frac{85}{3}$.

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