Home
Class 11
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 expression \( C_0 - 3C_1 + 5C_2 - \ldots + (-1)^n(2n + 1)C_n = 0 \), we will follow a systematic approach using the properties of binomial coefficients and summation. ### Step-by-Step Solution: **Step 1: Define the General Term** Let \( T_r \) be the general term of the series, which can be expressed as: \[ ...
Promotional Banner

Topper's Solved these Questions

  • Solutions of Triangle & Binomial Theorem

    ALLEN|Exercise EXERCISE-I|29 Videos

  • Solutions of Triangle & Binomial Theorem

    ALLEN|Exercise EXERCISE-II|11 Videos

  • Solutions of Triangle & Binomial Theorem

    ALLEN|Exercise Do yourself -6|4 Videos

  • SOLUTION AND PROPERTIES OF TRIANGLE

    ALLEN|Exercise All Questions|62 Videos

  • TRIGNOMETRIC RATIOS AND IDENTITIES

    ALLEN|Exercise All Questions|1 Videos

Similar Questions

Explore conceptually related problems

prove that :C_(0)^(2)+3C_(1)^(@)+5C_(2)^(2)+...+(2n+1)C_(n)^(2)=((n+1)2n!)/((n!)^(2))

C_(0)-3C_(1)+5c_(3)+....+(-1)^(n)(2n+1)C_(n) is equal to

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

Prove that C_(0)2^(2)C_(1)+3C_(2)4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)C_(n)=0 where C_(r)=nC_(r)

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 C_(0)^(2)+C_(1)^(2)+...C_(n)^(2)=(2n!)/(n!n!)

ALLEN-Solutions of Triangle & Binomial Theorem-Illustration
  1. Find numerically greatest term is the expansion of (3-5x)^11 "when " x...

    Text Solution

    |

  2. Given T(3) in the expansion of (1-3x)^6 has maximum numerical value .F...

    Text Solution

    |

  3. Prove that : ""^(25)C(10)+""^(24)C(10)+……..+""^(10)C(10)=""^(26)C(11)

    Text Solution

    |

  4. A student is allowed to select at most n books from a collection of (...

    Text Solution

    |

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

    Text Solution

    |

  6. If (1+x)^n=underset(r=0)overset(n)C(r)x^r then prove that C(1)^2+2.C(2...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  9. Prove that : ""^(n)C(0).""^(2n)C(n)-""^(n)C(1).""^(2n-2)Cn(n)+""^(n)...

    Text Solution

    |

  10. If (1+x)^n=C(0)C1c+C(2)x^2+…..+C(n)x^n then show that the sum of the p...

    Text Solution

    |

  11. If (1+x)^n=C(0)+C(1)x+C(2)x^2+….+C(n)x^n then prove that (SigmaSigma)...

    Text Solution

    |

  12. Find the coffiecient of x^2 y^3 z^4 w in the expansion of (x-y-z+w)^(...

    Text Solution

    |

  13. Find the total number of terms in the expansion of 1(1+x+y)^(10) and c...

    Text Solution

    |

  14. Find the coffiecient of x^5 in the expansion of (2-x+ 3x ^2)^6

    Text Solution

    |

  15. If (1+X+x^2)^n = Sigma(r=0)^(2n) a(r) x^r then prove that (a) a(r)=a...

    Text Solution

    |

  16. If (6sqrt(6)+14 )^(2n+1) = [N]+F and F=N -[N] , where [.] denotes grea...

    Text Solution

    |

  17. Find the last three digits in 11^(50)

    Text Solution

    |

  18. Prove that 2222^(5555)+5555^(2222) is divisible by 7

    Text Solution

    |

  19. If x is so small such that its square and digher powers may be neglect...

    Text Solution

    |

  20. The value of cube root of 1001 upto five decimal places is

    Text Solution

    |