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Question

The sum of the series 2C0+C1222+C2323+C3424++Cnn+12n+1 is equal to , where(Cr=nCr)

A
3n+11n1
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B
3n+11n+1
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C
3n+1+1n+1
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D
3n11n+1
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Solution

The correct option is B 3n+11n+1
We have,
2C0+C1222+C2323+C3424++Cnn+12n+1
=nr=01r+1 nCr2r+1
=1n+1nr=0n+1r+1 nCr2r+1
=1n+1nr=0 n+1Cr+12r+1

=1n+1n+1s=1 n+1Cs2s [Putting r+1=s]

=1n+1[(n+1s=0 n+1Cs2s) n+1C020]
=1n+1[(1+2)n+11]=3n+11n+1

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