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Question

If 2nr=0ar(x2)r=2nr=0br(x3)r and ak=1 for all kn, then bn is equal to

A
2(n+1)C2n+1
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B
2n+1Cn+1
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C
2n+1Cn+2
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D
None of these
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Solution

The correct option is B 2n+1Cn+1
Clearly, bn is the coefficient of (x3)n in the expression 2nr=0br(x3)r.

bn=coefficient of (x3)n in (2nr=0ar(x2)r) (1)

=coefficient of (x3)n in n1r=0ar(x2)r+2nr=nar(x2)r

(ak=1 for all kn)

=coefficient of (x3)n in 2nr=n(x2)r

=coefficient of (x3)n in [(x2)n{(x2)n+11(x2)1}]

=coefficient of (x3)n in ((x2)2n+1(x2)nx3)

=coefficient of (x3)n+1 in {(x2)2n+1(x2)n}

=coefficient of (x3)n+1 in (x2)2n+1

=coefficient of (x3)n+1 in [(x3)+1]2n+1

=coefficient of (x3)n+1 in {2n+1r=0 2n+1Cr (x3)r}

= 2n+1Cn+1

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