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Question

Assertion :H1,H2,H3,....Hn be n H.M between a and b then value of H1+aH1−a+Hn+bHn−b=2n Reason: H1=(n+1)abnb+a,Hn=(n+1)abna+b obtained by interchanging the numbers a & b.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
a,H1,H2,....Hn,b are in H.P
1a,1H1...1H2,1b are in A.P
d=abab(n+1)
So 1H1=t2=a+nb(n+1)ab H1=(n+1)aba+nb
H1a=(n+1)abnb+a .....(A)
& Hn=(n+1)ana+b by interchanging a & b
Hnb=(n+1)ana+b ....(B)

Now using componendo & dividendo on (A) & (B) and then adding, we get
H1+aH1a+Hn+bHnb=(2n+1)b+aba+(2n+1)a+bab
={(2n+1)b+a}{(2n+1)a+b}ba
=(2n+1)(ba)(ba)ba=(ba)[2n+11]ba
Therefore, H1+aH1a+Hn+bHnb=2n
Ans: A

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