Question

In an A.P. If $m\left(a_m\right)=n\left(a_n\right)$. Find $(a_{m+n}{)}^{th}$ term

Answer 1

$n$th term of an AP is $a+(n-1)d$

where $a$ is the first term and $d$ is the common difference

$a_m=a+(m-1)d$

$a_n=a+(n-1)d$

It is given that

$m\left(a_m\right)=n\left(a_n\right)$

$m\{a+(m-1\left)d\right\}=n\{a+(n-1\left)d\right\}$

$ma+m(m-1)d=na+n(n-1)d$

$ma-na=d(n^2-n-m^2+m)$

$a(m-n)=d(n-m)(m+n-1)$

$a=-d(m+n-1)$

Now $(m+n)$th term is

$a_{m+n}=a+(m+n-1)d$

$=-d(m+n-1)+(m+n-1)d$

$=0$