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Statement-1: sum(r =0)^(n) (r +1)""^(n)...

Statement-1: ` sum_(r =0)^(n) (r +1)""^(n)C_(r) = (n +2) 2^(n-1)`
Statement -2: ` sum_(r =0)^(n) (r+1) ""^(n)C_(r) x^(r) = (1 + x)^(n) + nx (1 + x)^(n-1)`

A

STATEMENT - 1 is True, STATEMENT - 2 is True and STATEMENT - 2 is correct explanation for STATEMENT - 1.

B

STATEMENT - 1 is True, STATEMENT - 2 is True and STATEMENT - 2 is NOT correct explanation for STATEMENT - 1.

C

STATEMENT-1 is True, STATEMENT-2 is False

D

STATEMENT-1 is False, STATEMENT-2 is True

Text Solution

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The correct Answer is:
A
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Stetemet - 1: sum_(r=0)^(n) r. ""^(n)C_(r) = n 2^(n-1) Statement-2: sum_(r=0)^(n) r. ""^(n)C_(r) x^(r) = n (1 + x )^(n-1) x

Statement-1 sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = nx (1 - x)^(n -1) Statement-2: sum_(r=0)^(n)r ""^(n)C_(r) x^(r) (-1)^(r) =0

Statement -1: sum_(r=0)^(n) r(""^(n)C_(r))^(2) = n (""^(2n -1)C_(n-1)) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))^(2)= ""^(2n)C_(n)

Statement -2: sum_(r=0)^(n) (-1)^( r) (""^(n)C_(r))/(r+1) = (1)/(n+1) Statement-2: sum_(r=0)^(n) (-1)^(r) (""^(n)C_(r))/(r+1) x^(r) = (1)/((n+1)x) { 1 - (1 - x)^(n+1)}

Statement-1: sum_(r=0)^(n) (1)/(r+1) ""^(n)C_(r) = (1)/((n+1)x) {( 1 + x)^(n+1) -1}^(-1) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))/(r+1) = (2^(n+1))/(n+1) .

Evaluate : sum_(r = 1)^(n) ""^(n)C_(r) 2^r

Statement-1 : sum_(r=0)^(n) r^(2) ""^(n)C_(r) x^(r) = n (n-1) x^(2) (1 + x)^(n-2) + nx (1 +x)^(n-1) Statement-2: sum_(r=0)^(n) r^(2) ""^(n)C_(r) = n (n-1)2^(n-2)+ n2^(n-1) .

RESONANCE-DPP-QUESTION
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