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asked in Mathematics by (269 points)

If an+1 + bn+1/an + bn is the A.M. between a and b. Then, find the value of n.

1 Answer

+1 vote
answered by (33.5k points)
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We know the arithmetic mean of ‘a’ and ‘b’ is (a+b)/2
Hence (a+b)/2 = (an+1+bn+1)/(an+bn)
(a+b)/2 = (an x a + bn x b)/(an+bn)  
(ana + anb + bna + bnb) = (2ana+2bnb)
(anb + bna) = (ana + bnb)
an(b – a)=bn(b – a)
an = bn
This is possible only when n = 0

commented by (269 points)
Won't it be possible if n=2 in the last step?
commented by (33.5k points)
No, as 'a' and 'b' are variables and won't be equal if n=2
If we take n=0 then a^0 becomes 1 and b^0 becomes 0 and hence we get LHS=RHS
commented by (269 points)
Thank you sir.

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