Home
Class 12
MATHS
Prove that :C(0)-3C(1)+5C(2)- ………..(-1)^...

Prove that :`C_(0)-3C_(1)+5C_(2)- ………..(-1)^n(2n+1)C_(n)=0`

Text Solution

AI Generated Solution

To prove the statement \( C_0 - 3C_1 + 5C_2 - \ldots + (-1)^n(2n+1)C_n = 0 \), we will follow a systematic approach using the properties of binomial coefficients and summation. ### Step-by-Step Solution: 1. **Understanding the Expression**: The expression can be rewritten in summation notation: \[ S = \sum_{r=0}^{n} (-1)^r (2r + 1) C_r ...
Promotional Banner

Topper's Solved these Questions

  • BIONOMIAL THEOREM

    ARIHANT MATHS ENGLISH|Exercise JEE Type Solved Example : (Matching Type Questions )|2 Videos

  • BIONOMIAL THEOREM

    ARIHANT MATHS ENGLISH|Exercise Exercise For Session 1|7 Videos

  • AREA OF BOUNDED REGIONS

    ARIHANT MATHS ENGLISH|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|23 Videos

  • CIRCLE

    ARIHANT MATHS ENGLISH|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|16 Videos

Similar Questions

Explore conceptually related problems

(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - 2C_(1) + 3C_(2) - 4C_(3) + … + (-1)^(n) (n+1) C_(n) = 0

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - (C_(1))/(2) + (C_(2))/(3) -…+ (-1)^(n) (C_(n))/(n+1) = (1)/(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_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) - C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

Prove that .^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3) + "……" + (-1)^(r) + .^(n)C_(r) + "……" = (-1)^(r ) xx .^(n-1)C_(r ) .

If C_(0),C_(1),C_(2),C_(3), . . .,C_(n) be binomial coefficients in the expansion of (1+x)^(n) , then Q. The value of the expression C_(0)-2C_(1)+3C_(2)-. . . .+(-1)^(n)(n+1)C_(n) is equal to

If C_(0),C_(1), C_(2),...,C_(n) denote the cefficients in the expansion of (1 + x)^(n) , then C_(0) + 3 .C_(1) + 5 . C_(2)+ ...+ (2n + 1) C_(n) = .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) C_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - …+ (-1)^(n) C_(n) C_(0) = 0 or (-1)^(n//2) (n!)/((n//2)!(n//2)!) , according as n is odd or even .

If C_(0) , C_(1), C_(2), …, C_(n) are the binomial coefficients in the expansion of (1 + x)^(n) , prove that (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1)) ((n-2)^(n))/((n+1)!) prod _(r=1)^(n) (C_(r-1) + C_(r)) .

"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)

ARIHANT MATHS ENGLISH-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Prove that :C(0)-3C(1)+5C(2)- ………..(-1)^n(2n+1)C(n)=0

    Text Solution

    |

  2. The value of ((30), (0))((30), (10))-((30), (1))((30),( 11)) +(30 2)(3...

    Text Solution

    |

  3. If the coefficient of the rth, (r+1)th and (r+2)th terms in the expans...

    Text Solution

    |

  4. If the coefficient of x^(7)in [ax^(2) + (1/bx)]^(11) equals the coeffi...

    Text Solution

    |

  5. For natural numbers m ,n ,if(1-y)^m(1+y)^n=1+a1y+a2y^2+... , and a1=a2...

    Text Solution

    |

  6. In the binomial expansion of (a - b)^n , n ge 5 the sum of the 5th ...

    Text Solution

    |

  7. The sum of series ^^(20)C0-^^(20)C1+^^(20)C2-^^(20)C3++^^(20)C 10 is 1...

    Text Solution

    |

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

    Text Solution

    |

  9. The reamainder left out when 8^(2n) - (62)^(2n+1) is divided by 9 is

    Text Solution

    |

  10. For r = 0, 1,"…..",10, let A(r),B(r), and C(r) denote, respectively, t...

    Text Solution

    |

  11. Let S(1) = sum(j=1)^(10) j(j-1).""^(10)C(j), S(2) = sum(j=1)^(10)j."...

    Text Solution

    |

  12. Find the coefficient of x^7 in the expansion of (1 - x -x^2 + x^3)^(6)...

    Text Solution

    |

  13. If n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is ...

    Text Solution

    |

  14. The term independent of x in expansion of ((x+1)/(x^(2/3)-x^(1/3)+1)-(...

    Text Solution

    |

  15. The coefficients of three consecutive terms of (1+x)^(n+5) are in the ...

    Text Solution

    |

  16. If the coefficient of x^(3) and x^(4) in the expansion of (1+ax+bx^(2)...

    Text Solution

    |

  17. Coefficient of x^(11) in the expansion of (1+x^2)(1+x^3)^7(1+x^4)^(12)...

    Text Solution

    |

  18. The sum of coefficient of integral powers of x in the binomial expansi...

    Text Solution

    |

  19. The coefficient of x^9 in the expansion of (1+x)(16 x^2)(1+x^3)(1+x^(1...

    Text Solution

    |

  20. If the number of terms in the expansion of (1-2/x+4/(x^(2))) x ne 0, i...

    Text Solution

    |

  21. Let m be the smallest positive integer such that the coefficient of x^...

    Text Solution

    |