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 $S_n$ , if $a = 5, d = 3$ and $a_n = 50$.

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 $S_n$ , if $a = 5, d = 3$ and $a_n = 50$.

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,

$a_n=5+(n-1)3$

$50=5+(n-1)3$

$50-5=(n-1)3$

$45=(n-1)3$

$n-1=15$

$n=15+1$

$n=16$

$S_n=\frac{16}{2}[2 \times 5+(16-1) \times 3]$

$=8[10+15 \times 3]$

$=8(10+45)$

$=8 \times 55$

$=440$

Therefore, $n=16$ and $S_n=440$.