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

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NCERT Maths solution class 10 chapter 5 arithmetic progressions exercise 5.2 question number 12,this is a very common question student face problem in solving. Here Two A.P’s have same common difference (d) and  difference of their 100th term is given to us as 100, you need to find the difference between their 1000th term.

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1. let, the first term of AP 1 be a1 and the first term of AP 2 is a2,

because common difference is given same for both the AP’s

∴ let the common difference of both AP’s be d.

to find the nth tern we have a formula,

an = a+(n−1)d

Therefore, For the first A.P.,we know,

a100 = a1+(100−1)d

a1 + 99d

&

a1000 = a1+(1000−1)d

a1000 = a1+999d

similarly,For second A.P.,

a100 = a2+(100−1)d

a2+99d

&

a1000 = a2+(1000−1)d

a2+999d

It is Given that, difference between 100th term of the two APs = 100

Therefore, (a1+99d) − (a2-99d) = 100Therefore,

a1+99da2-99d = 100

a1a2 = 100……………………………………………………………….. (i)

Difference between 1000th terms of the two AP’s is

(a1+999d) − (a2+999d) = a1a2

From equation (i),

This difference, a1a= 100

Hence, the difference between 1000th terms of the two A.P. will be 100.

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