Q : 12       Two APs have the same common difference. The difference between their   $\small 100$th terms is  $\small 100$, what is the difference between their $\small 1000$th terms?

G Gautam harsolia

It is given that
Two APs have the same common difference and  difference between their   $\small 100$th terms is  $\small 100$
i.e.
$a_{100}-a'_{100}= 100$
Let common difference of both the AP's be d
$\Rightarrow a+99d-a'-99d=100$
$\Rightarrow a-a'=100 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - (i)$
Now, difference between 1000th term is
$a_{1000}-a'_{1000}$
$\Rightarrow a+999d -a'-999d$
$\Rightarrow a-a'$
$\Rightarrow 100 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (using \ (i) )$
Therefore, difference between 1000th term is 100

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