Question

# Statement-1: ∑nr=0(r+1)nCr=(n+2)2n−1. Statement-2: ∑nr=0(r+1)nCrxr=(1+x)n+nx(1+x)n−1.

A
Statemet -1 is false, Statement - 2 is ture
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B
Statemet -1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
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C
Statement -1 is true, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1
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D
Statement -1 is true, Statement -2 is false
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Solution

## The correct option is A Statemet -1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1We have ∑nr=0(r+1)nCrxr=∑nr=0rnCrxr+∑nr=0nCrxr=∑nr=1 r.nrn−1Cr−1xr+(1+x)n=nx∑nr=1n−1Cr−1xr−1+(1+x)n=nx(1+x)n−1+(1+x)n=RHS ∴ Statement 2 is correct. Putting x=1, we get ∑nr=0(r+1)ncr=n.2n−1+2n=(n+2).2n−1 ∴ Statement 1 is also true and statement 2 is a correct explanation for statement 1.

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