Question

For what value of $n,$ the $n^{th}$ term of A.P. $63,65,67,........$ and $n^{th}$ term of A.P. $3,10,17,..........$ are equal to each other?

Answer 1

First A.P: $63,65,67,........$

First term, $a=63$

Common difference, $d=a_2-a_1=65-63=2$

We know that the $n^{th}$ term of an AP is given by $a_n=a+(n-1)d$

For the first AP, we have

$n^{th}$ term, $a_n=63+(n-1)2$

$=63+2n-2$

$=2n+61$

Second A.P: $3,10,17,..........$

Here, First term $b=3,$ and common difference, $d=b_2-b_1=10-3=7$

For the second AP, we have

$n^{th}$ term, $b_n=3+(n-1)7$

$=3+7n-7$

$=7n-4$

Let the $n^{th}$ terms are equal for both the APs.

i.e, $7n-4=2n+61$

$\Rightarrow 5n=65$

$\Rightarrow n=\frac{65}{5}=13$

Thus, $n=13$

Hence, the ${13}^{th}$ term of both the APs will be equal.