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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? Q.12

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What is the best solution for exercise 5.2 of math in arithmetic progressions of class 10th. It is very important type of question. How i solve easily this question 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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1. Let, the first term of two APs beÂ a1Â andÂ a2Â respectively

And the common difference of these APs beÂ d.

For the first A.P.,we know,

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 second A.P., we know,

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

Given that, difference between 100thÂ term of the two APs = 100

Therefore, (a1+99d) âˆ’ (a2+99d) = 100

a1âˆ’a2Â = 100â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..Â (i)

Difference between 1000thÂ terms of the two APs

(a1+999d) âˆ’ (a2+999d) =Â a1âˆ’a2

From equationÂ (i),

This difference,Â a1âˆ’a2Â = 100

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

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