Question

For what value of $n$ are the $n^{th}$ terms of the following two A.P's the same?
(i) $1,7,13,19,....$ (ii) $69,68,67,....$

Answer 1

Clearly, $1,7,13,19,....$ forms an A.P. with first term $1$ and common difference $6$. Therefore, its $n^{th}$ term is given by

$a_n=1+(n-1)\times 6=6n-5$
Also, $69,68,67,....$ forms an A.P. with first $69$ and common difference $-1$.

So, its $n^{th}$ term is given by

$a_n^{\prime }=69+(n-1)\times (-1)=-n+70$
The two A.P's will have identical $n^{th}$ terms, if
$a_n=a_n^{\prime }$

$⟹6n-5=-n+70$

$⟹7n=75$

$⟹n=\frac{75}{7}$ which is not a natural number.

Hence, there is no value of $n$ for which two A.P's will have identical terms.