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Question

Consider the binomial expansion of (x+124x)n,nN where the terms of the expansion are written in decreasing powers of x. If the coefficients of the first three terms form an arithmetic progression, then which of the following statements hold(s) good?

A
Total number of terms in the binomial expansion is 8
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B
Number of terms in the binomial expansion with integral power of x is 3
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C
There is no term in the binomial expansion which is independent of x
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D
Fourth and fifth terms are the middle terms of the expansion
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Solution

The correct option is C There is no term in the binomial expansion which is independent of x
(x1/2+12x1/4)n
Tr+1=nCr x(nr)/212rxr/4
Coefficient of the first three terms are
nC0, nC112, nC2122nC0+nC214=2nC1121+n(n1)8=nn(n1)8=n1n=8 (n1)

Tr+1=nCr x(nr)/212rxr/4 =8Cr12rx4 3r4

Hence, there are total 9 terms in the binomial expansion.
And 5th term is the middle term.

Terms with integer power occur when r=0,4,8
Hence, 3 such terms are there.

For being independent of x:
43r4=0
r=163 (not possible)
Hence, no such term is there.

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