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# If two A.P have the same common difference. The difference b/w their 100th term is 100. What is the difference between their 1000th term.

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The question is given from NCERT Book of class 10th Chapter no. 5 Ex. 5.2 Q. 12. In the following question you have to find the difference between the 1000th term as said in the above question. Give the solution of the above question.

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1. Solution:

Let, the 1st term of two AP’s be a1Â andÂ a2 ;

The c.d of the AP’s be ” d ” .

For 1st A.P

anÂ =Â a+(nâˆ’1)d

Therefore,

a100Â =Â a1+(100âˆ’1)d

=Â a1Â + 99d

a1000Â =Â a1+(1000âˆ’1)d

a1000Â =Â a1+999d

For 2nd A.P ,

anÂ =Â a+(nâˆ’1)d

Therefore,

a100Â =Â a2 + (100âˆ’1)d

=Â a2 + 99d

a1000Â =Â a2 + (1000âˆ’1)d

=Â a2 + 999d

It is given that the difference between 100thÂ term of the two APs = 100

i,e (a1 + 99d) âˆ’ (a2 + 99d) = 100

a1âˆ’a2 = 100Â  Â  Â  Â  Â â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (i)

Diff. b/w 1000th terms of the two APs

(a1+999d) âˆ’ (a2+999d) =Â a1âˆ’a2

From eq. (i),

This diff.Â  , a1âˆ’a2Â = 100

The difference between 1000th terms of the 2 A.P’s will be 100.

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