Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?

Given:

Two arithmetic progressions have the same common difference. The difference between their 100th terms is 100.

To do: We have to find the  difference between their 1000th terms.

Solution:

Let $a, a+d, a+2d,......$ and $p, p+d, p+2d,.......$ be the two A.P.s.

Therefore,

$a_{100}=a+(100-1)d$

$=a+99d$

$p_{100}=p+(100-1)d$

$=p+99d$

According to the question,

$a+99d-(p+99d)=100$

$a-p=100$.....(i)

$a_{1000}=a+(1000-1)d$

$=a+999d$

$p_{1000}=p+(1000-1)d$

$=p+999d$

Therefore,

$a_{1000}-p_{1000}=a+999d-(p+999d)$

$=a+999d-p-999d$

$=a-p$

$=100$     (From (i))

The difference between their 1000th terms is $100$. 

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Updated on: 10-Oct-2022

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