What is the common difference of an A.P. in which $a_{21} -a_{7} = 84?$

Given: An A.P. in which $a_{21} -a_{7} =\ 84$

To do: to find he common difference of an A.P.

Let a be the first term and d is the common difference of the given A.P.

As known $n^{th}$ term of and A.P., $a_{n} =a+( n-1) d$

$a_{21} =a+( 21-1) d$

$\Rightarrow a_{21} =a+20d\ \ \ \ \ \ ..........( 1)$

$a_{7} =a+( 7-1) d$

$\Rightarrow a_{7} =a+6d\ \ \ \ \ \ ...........( 2)$

On subtracting $( 2)$ from $( 1)$

we have, $a_{21} -a_{7} = a+20d-a-6d=14d$

$\Rightarrow 14d=84\ \ \ \ \ \ \ ( a_{21} -a_{7} =\ 84)$

$\Rightarrow d=\frac{84}{14} =6$

Thus the common difference of the given A.P. is $6$.