Let there be an A.P. with first term ‘$a$’, common difference '$d$'. If $a_n$ denotes its $n^{th}$ term and $S_n$ the sum of first $n$ terms, find.$n$ and $a_n$, if $a = 2, d = 8$ and $S_n = 90$.
Given:
In an A.P., first term $=a$ and common difference $=d$.
$a_n$ denotes its $n^{th}$ term and $S_n$ the sum of first $n$ terms.
To do:
We have to find $n$ and $a_n$, if $a = 2, d = 8$ and $S_n = 90$.
Solution:
We know that,
$\mathrm{S}_{n}=\frac{n}{2}[2 a+(n-1) d]$
$n$th term $a_n=a+(n-1)d$
This implies,
$a_n=2+(n-1)8$
$=2+8n-8$
$=8n-6$........(i)
$S_n=\frac{n}{2}[2 \times 2+(n-1)8]$
$90=\frac{n}{2}[4+8n-8]$ (From (i))
$90(2)=n(8n-4)$
$180=4n(2n-1)$
$n(2n-1)=45$
$2n^2-n-45=0$
$2n^2-10n+9n-45=0$
$2n(n-5)+9(n-5)=0$
$(2n+9)(n-5)=0$
$n=5$ or $2n=-9$ which is not possible as $n$ cannot be negative
$\therefore n=5$
This implies,
$a_n=8(5)-6$
$=40-6$
$=34$
Therefore, $n=5$ and $a_n=34$.  
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