Home
Class 12
MATHS
If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n ...

If `(1+x)^n=C_0+C_1x+C_2x^2+C_3x^3+...+C_nx^n` then prove that `2.C_0+2^2C_1/2+2^3C_2/3+2^4C_3/4+...+2^(n+1)C_n/(n+1)=(3^(n+1)-1)/(n+1)`

Text Solution

AI Generated Solution

Promotional Banner

Topper's Solved these Questions

  • Binomial Theorem for Positive Integrel Index

    A DAS GUPTA|Exercise Exercise|113 Videos

  • Application of dy/dx

    A DAS GUPTA|Exercise Exercise|45 Videos

  • Circles

    A DAS GUPTA|Exercise EXERCISE|122 Videos

Similar Questions

Explore conceptually related problems

If (1+x)^n=C_0+C_1x+C_2x^2+C_3x^3+...+C_nx^n then prove that C_0-1/2C_1+1/3C_2-1/4C_3+...+(-1)^n*C_n/(n+1)=1/(n+1)

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + (C_1)/(2) + (C_2)/(3) + ……. + (C_n)/(n+1) = (2^(n+1) -1)/(n+1)

2C_0 + 2^2 (C_1)/(2) + 2^3 (C_2)/(3) + ………. + 2^(n+1) (C_n)/(n+1) = (3^(n+1) - 1)/(n+1)

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + 2C_1 + ….. + 2 ""^nC_n = 3^n

If (1+ x)^(n) = C_(0) + C_(1) x + C_(2)x^(2) + ...+ C_(n)x^(n) , prove that C_(1) + 2C_(2) + 3C_(3) + ...+ n""C_(n) = n*2^(n-1)

If (1+x)^n=C_0+C_1x+C_2x^2+……..+C_nx^n , show that 3.C_0+3^2.C_1/2+3^3.C_2/2+.+3^(n+1). C_n/(n+1)=(4^(n+1)-1)/(n+1)

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+...+C_(n)x^(n) then 2C_(0)+(C_(1))/(2)+2^(3)(C_(2))/(3)+...+2^(n+1)(C_(n))/(n+1)=

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) +… + C_(n) x^(n) , prove that C_(0) + 2C_(1) + 3C_(2) + …+ (n+1)C_(n) = (n+2)2^(n-1) .

If (1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+C_(3).x^(3)+......+C_(n).x^(n), then prove that C_(0)+2C_(1)+4C_(2)+6C_(3)+...+2n.C_(n)=1+n*2^(n)

If (1+x)^n=C_0+C_1x+C_2x^2+_____+C_nx^n , prove that C_1+2C_2+3C_3+_____+ ^nC_n=n2^(n-1)

A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Prove by binomial expansion that sum(k=1)^nk^2*"^nCk=n(n+1)2^(n-2)

    Text Solution

    |

  2. Evaluate sum(r=0)^n(r+1)^2*"^nCr

    Text Solution

    |

  3. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that 2.C0+2^2C1/2+2...

    Text Solution

    |

  4. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that C0-1/2C1+1/3C2...

    Text Solution

    |

  5. Prove that 3*""^10C0+3^2*(""^10C1)/2+3^3*(""^10C2)/3+...3^11*(""^10C10...

    Text Solution

    |

  6. Prove that 2*""^nC0+2^2*(""^nC1)/2+2^3*(""^nC2)/3+...2^(n+1)*(""^nCn)/...

    Text Solution

    |

  7. Find the sum sum(k=0)^n("^nCk)/((k+1)(k+2))

    Text Solution

    |

  8. Find the sum sum(r=0)^n(-1)^r*(""^nCr)/(""^(r+3)Cr)

    Text Solution

    |

  9. Find the sum sum(k=1) (""^nC(2k-1))/(2k)

    Text Solution

    |

  10. Find the sum sum(k=0)^n ("^nCk)/(k+1)

    Text Solution

    |

  11. If (1+x)^n=sum(r=0)^n Crx^r then prove that sum(r=0)^n (Cr)/((r+1)2^(r...

    Text Solution

    |

  12. Find sum(r=0)^n(r+1)*"^nCrx^r

    Text Solution

    |

  13. Show that C0^2-C1^2+C2^2-C3^2+...........+(-1)^n Cn^2=0 or (-1)^(n/2)C...

    Text Solution

    |

  14. Prove that (""^(2n)C(0))^(2)-(""^(2n)C(1))^(2)+(""^(2n)C(2))^(2)-…+(""...

    Text Solution

    |

  15. Sum of the products of the binomial coefficients C0,C1,C2,......Cn ta...

    Text Solution

    |

  16. Find the sum ^20C10.^15C0+^20C9.^15C1+^20C8.^15C2+....+^20C0.^15C10

    Text Solution

    |

  17. Prove that sum(r=1)^k (-3)^(r-1) "^(3n)C(2r-1) =0 , where k=(3n)/2 and...

    Text Solution

    |

  18. If p+q=1, then show that sum(r=0)^n r^2^n Crp^r q^(n-r)=n p q+n^2p^2do...

    Text Solution

    |

  19. Use a combinatorial argument to prove that (C(n,1))^2+2(C(n,2))^2+3(C...

    Text Solution

    |

  20. Prove that (""^nC1)/(""^nC0)+2*(""^nC2)/(""^nC1)+3*(""^nC3)/(""^nC2)+....

    Text Solution

    |