Question

The sum of $5^{th}$ and $9^{th}$ terms of an $A.P.$ is $72$ and the sum of $7^{th}$ and ${12}^{th}$ terms is $97$. Find the
(a) ${25}^{th}$ term of an $AP$
(b) sum of first $50$ terms of the $AP$

Answer 1

$A_5+a_9=72$

$a+4d+a+8d=72$

$2a+12d=72$............eq1

and

$a_7+a_{12}=97$

$a+6d+a+11d=97$

$2a+17d=97$.............eq2

subtracting eq1 from eq2

$5d=25$

$d=5$

put $d=5$ in eq1

$2a+12\left(5\right)=72$

$2a+60=72$

$2a=72-60$

$a=6$

$T_{25}=a+(n-1)d$

$=6+(25-1)5=126$

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

$=\frac{50}{2}\left[2\right(6)+(50-1\left)5\right]$

$=25[12+49\times 5]=6425$