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Prove that (i) C(1)+2C(2)+3C(3)+……+nC(...

Prove that (i) `C_(1)+2C_(2)+3C_(3)+……+nC_(n)=n.2^(n+1)`
(ii) `C_(0)+(C_(1)/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)`

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MOTION-BINOMIAL THEOREM -Exercise -3 ( Subjective )
  1. (C(0)+C(1))(C(1)+C(2))(C(2)+C(3))...(C(n-1)+C(n))=(C(0).C(1).C(2)........

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  2. If P(n) denotes the product of all the coefficients in the expansion o...

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  3. Prove that (i) C(1)+2C(2)+3C(3)+……+nC(n)=n.2^(n+1) (ii) C(0)+(C(1)...

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  4. 2.C(0)+(2^(2).C(1))/(2)+(2^(3).C(2))/(3)+(2^(4).C(3))/(4)+......+(2^(n...

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  5. If (1 + x)^(n) = C(0) + C(1) x + C(2) x^(2) + C(3) x^(3) + … + C(n) x...

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  6. (1 + x)^(n) = C(0) + C(1) x + C(2) x^(2) + C(3) x^(3) + … + C(n) x^(n)...

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  7. prove that : C0^2+3C1^@+5C2^2+...+(2n+1)Cn^2=((n+1)2n!)/(n!)^2

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  8. If (1 + x)^(n) = C(0) = C(1) x + C(2) x^(2) + …+ C(n) x^(n) , find...

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  9. Prove that sqrt(C1) +sqrt(C2) +sqrt(C3)+...+sqrt(Cn)le 2^(n-1) +(n-1)/...

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  10. If C(r ) = (n!)/(r!(n - r)!), then prove that sqrt(C(1)) + sqrt(C(2)...

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  11. Sum of all the digits of the coefficient of x^(5) in the expansion of ...

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  12. Find the coefficient of x^(4) in the expansion of (1+x+x^(2)+x^(3))^(1...

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  13. Find the coefficients of x^(4) in the expansions of (2 - x + 3x^(2...

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  14. Let a = (72!)/(36!)^2-1 , then (A) a is odd (B) a is divisible by 7...

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  15. find the sum of the series sum(r=0)^n(-1)^r *"^nCr[1/(2^r)+(3^r)/(2^(...

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  16. Given that (1+x+x^2)^n=a0+a1x+a2x^2+........+a(2n)x^(2n) find the val...

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  17. Given that (1+x+x^(2))^(n)=a(0)+a(1)x+a(2)x^(2)+......+a(2n)x^(2n), f...

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  18. Given that (1+x+x^(2))^(n)=a(0)+a(1)x+a(2)x^(2)+......+a(2n)x^(2n), f...

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  19. If n(j(r))=((1-xn)(1-x(n-1))(1-x^(n-2)).......(1-x^(n-r+1)))/((1-x)(1-...

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  20. Prove that sum(k=0)^n ^nCk sinKx.cos(n-K)x=2^(n-1)sin nx

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