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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)`

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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) .

ML KHANNA-BINOMIAL THEOREM AND MATHEMATICAL INDUCTION -Problem Set (4) (MULTIPLE CHOICE QUESTIONS)
  1. If C(0), C(1), C(2), ..., C(n) denote the binomial cefficients in t...

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  2. If C(0), C(1), C(2), …. C(n) denote the coefficients in the expansion ...

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  3. Statement-1: sum(r =0)^(n) (r +1)""^(n)C(r) = (n +2) 2^(n-1) Stat...

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  4. If (1+x)^(n)=C(0)+C(1)x+C(2)x^(2)+…+C(n)x^(n), then C(0)+5C(1)+9C(2)+...

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  5. If (1+x)^(n)= sum(r=0)^(n)C(r )x^(r ) and sum(r =0)^(n) (C(r ))/(r+1)=...

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  6. If (1+x)^(n)=sum(r=0)^(n)*C(r )x^(r ) and sum(r=0)^(n) (-1)^(r ) (C(r ...

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  7. The sum of the series .^(20)C(0)-.^(20)C(1)+ .^(20)C(2)-.^(20)C(3)+...

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  8. The sum to (n+1) terms of the series (C(0))/(2)-(C(1))/(3)+(C(2))/(...

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  9. If C(r ) stands for ""^(n)C(r ), then the sum of first (n+1) terms of...

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  10. The value of the sum of the series 3.""^(n)C(0)-8" "^(n)C(1)+13" "^(n...

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  11. If (1+x)^(n) = C(0)+C(1)x + C(2) x^(2) +...+C(n)x^(n) then C(0)""^...

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  12. If n is a positive integer and C(k)=""^(n)C(k), then the value of sum(...

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  13. The vaule of sum(r=0)^(n-1) (""^(C(r))/(""^(n)C(r) + ""^(n)C(r +1)) ...

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  14. sum(r=0)^(n) (-1)^(r )" "^(n)C(r ) (1+r x)/(1+n x) equals

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  15. If n gt 3, then abC(0)-(a-1) (b-1) C(1) + (a-2)(b-2) C(2)-(a-3) (b-3) ...

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  16. If C(r) be the coefficients of x^(r) in (1 + x)^(n) , then the value ...

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  17. If n is an odd natural number, then sum(r=0)^(n) ((-1)^(r ))/(""^(n)C(...

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  18. If a(n) = sum(r=0)^(n) (1)/(""^(n)C(r)) , find the value of sum(...

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  19. With usual notations C(0)C(1)+C(1)C(2)+…+C(n-1)C(n)=

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  20. With usual notations C(0)C(2)+C(1)C(3)+C(2)C(4)+…+C(n-2)C(n)=

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