Question

The 13th term of an A.P. is four times its 3rd term. if its fifth term is 16, then find the sum of its first ten terms

Answer 1

Since,

$T\left(13\right)=4\times T\left(3\right)$

$T\left(5\right)=16$

We know that $a\left(n\right)=a+(n-1)d$

so

$a\left(13\right)=a+12d$

$a\left(3\right)=a+2d$

By solving this we get

$a+12d=4a+8d$

$a=\frac{4}{3}\times d$

also,

$a\left(5\right)=a+4d=16$

putting value of a we get,

$d=3$

also

$a=4$

as we know,

sum $\left(n\right)=(\frac{n}{2}\times (a\left(1\right)+a\left(n\right))$

$a\left(10\right)=4+(9\times 3)=31$

sum$\left(10\right)=5\times (4+31)=175$

This is the required answer....