Write the expression $a_n – a_k$ for the A.P. $a, a + d, a + 2d, ……$Hence, find the common difference of the A.P. for which$a_{10} – a_5 = 200$
Given:
Given A.P. is $a, a + d, a + 2d, ……$
$a_{10} – a_5 = 200$
To do:
We have to find $a_{n} - a_{k}$ and the common difference of the A.P.
Solution:
$a_1=a, a_2=a+d, a_3=a+2d$ and $d=a_2-a_1=a+d-(a)=a+d-a=d$
nth term of the A.P. $a_n=a+(n-1)d$
kth term of the A.P. $a_k=a+(k-1)d$
$a_n-a_k=a+(n-1)d-[a+(k-1)d]$
$=a+nd-d-a-kd+d$
$=(n-k)d$
According to the question,
$a_{10} – a_5 = 200$
$200=a+(10-1)d-(a+(5-1)d)$
$200=a+9d-a-4d$
$200=5d$
$d=\frac{200}{5}$
$d=40$
Hence, $a_{n}-a_{k}$ is $(n-k)d$ and the common difference is $40$.  
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