In an AP:
Given $a = 8, a_n = 62, S_n = 210$, find $n$ and $d$.
Given:
In an A.P., $a = 8, a_n = 62, S_n = 210$
To do:
We have to find $n$ and $d$.
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,
$a_n=8+(n-1)d$
$62=8+(n-1)d$
$62-8=(n-1)d$
$54=(n-1)d$
$(n-1)d=54$........(i)
$S_n=\frac{n}{2}[2 \times 8+(n-1)d]$
$210=\frac{n}{2}[16+54]$ (From (i))
$210(2)=n(70)$
$3(2)=n$
$n=6$
$\therefore (6-1)d=54$
$5d=54$
$d=\frac{54}{5}$
Therefore, $n=6$ and $d=\frac{54}{5}$.
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