Question

For what value of $n$, are the $nth$ terms of two $APs:63,65,67,...$ and $3,10,17$, .... equal?

Answer 1

First A.P. is $63,65,67,\dots$

$\therefore a=63$

$d=65-63=2$

$n^{th}$ term of this A.P. $=a_n=a+(n-1)d$

$a_n=63+(n-1)2=63+2n-2$

$a_n=61+2n$……………….(1)

Second A.P is $3,10,17,\dots$

$a=3$

$d=10-3=7$

$n^{th}$ term of this A.P. $=3+(n-1)7$

$a_n=3+7n-7$

$a_n=7n-4$………………..(2)

It is given that, $n^{th}$ term of these A.P.s are equal to each other.

Equating equations (1) and (2),

$61+2n=7n-4$

$61+4=5n$

$5n=65$

$n=13$

Therefore, ${13}^{th}$ terms of both these A.P.s are equal to each other.