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Question

If the first and the (2n+1)th terms of an A.P., a G.P. and a H.P. of positive terms are same and their (n+1)th terms are a,b and c, respectively, then

A
a=b=c
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B
abc
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C
a+c=2b
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D
ac=b2
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Solution

The correct options are
B abc
D ac=b2
If a sequence has 2n+1 terms in total, then the n+1th term is the middle term.
Let the 1st term is a1 and the 2n+1th term is a2n+1

For A.P. an+1=a1+a2n+12
a=a1+a2n+12(1)
Similarly for G.P. a1.a2n+1=an+1
b=a1.a2n+1(2)

Finally for H.P. an+1=2a1.a2n+1a1+a2n+1
c=2a1.a2n+1a1+a2n+1(3)
From (1),(2),(3) we get
b2=ac
and abc ( A.M.G.M.H.M.)

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