Home
Class 12
MATHS
Find the sum of the following C1 - 2C2 +...

Find the sum of the following `C_1 - 2C_2 + 3C_3 - ..... + n(-1)^(n-1) C_n`

Text Solution

Verified by Experts

L.H.S. : `C_0 + 3C_1 + 5C_2 + ... + (2n+1)C_n`
`sum_(k=1)^n (2k+1) C_k` = `C_0 + sum_(k=1)^n(2k+1) "^nC_k
`"^nC_0 + sum(k=1)^n 2k ^nC_k + sum(k=1)^n ^nC_k
`"^nC_0 + 2sum_(k=1)^n n ^(n-1)C_(k-1) + (^nC_1 + ^nC_2 + ^nC_n)`
`("^nC_0 + ^nC_1 + ^nC_2 + .... + ^nC_n) + 2n^(n-1)C_0 + ^(n-1)C_1 + ... + ^(n-1)C_(n-1) )
`2^n + 2n.2(n-1)` = `2^n + n.2^n` = `(n+1)2^n`
R.H.S.
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIAL EQUATIONS

    MBD PUBLICATION|Exercise QUESTION BANK|87 Videos

  • INTEGRATION

    MBD PUBLICATION|Exercise QUESTION BANK|418 Videos

Similar Questions

Explore conceptually related problems

Find the sum of the following C_1 + 2^2 C_2 + 3^2 C_3 + ... + n^2C_n

Find the sum of the following 1.2 C_(2) + 2.3 C_(3) + ... + (n-1)nCn

Find the sum of C_1 + 2C_2 + 3C_3 + .... + nC_n

Find the sum of C_0 + 2C_1 + 3C_2 + .... + (n+1)C_n

prove that :- C_1 - 1/2 C_2 + 1/3 C_3 + ... + (-1)^(n+1) 1/n C_n = 1+1/2 + .... +1/n

Show that C_2 + 2C_3 + 3C_4 + ... + (n-1)C_n = 1+ (n-2) 2^(n-1)

Show that 3C_0-8C_1 + 13C_2 - 18C_3 + ..... + (n+1)^(th) term = 0

Prove that "^(2n)C_1 + ^(2n)C_3 + .... + ^(2n)C_(2n-1) = 2^(2n-1)

Show that C_0 C_1 + C_1 C_2 + C_2 C_3 + .... + C_(n-1) C_n = (2n!)/((n-1)!(n+1!))

Prove the following by induction. 1 + 2+ 3+….+ n = n(n+1)//2

MBD PUBLICATION-Elements of Mathematics-QUESTION BANK
  1. Prove that "^(2n)C1 + ^(2n)C3 + .... + ^(2n)C(2n-1) = 2^(2n-1)

    Text Solution

    |

  2. Find the sum of C1 + 2C2 + 3C3 + .... + nCn

    Text Solution

    |

  3. Find the sum of C0 + 2C1 + 3C2 + .... + (n+1)Cn

    Text Solution

    |

  4. Compute ((1+k)(1+k/2) ..... (1+k/n))/((1+n)(1+n/2) ..... (1+n/k))

    Text Solution

    |

  5. Show that C0 C1 + C1 C2 + C2 C3 + .... + C(n-1) Cn = (2n!)/((n-1)!(n...

    Text Solution

    |

  6. C0 C1 + C1 C2 + .... + C(n-1) Cn = (2^n.n.1.3.5... (2n-1))/(n+1)

    Text Solution

    |

  7. Show that 3C0-8C1 + 13C2 - 18C3 + ..... + (n+1)^(th) term = 0

    Text Solution

    |

  8. Show that C0 n^2 + C1 (2-n)^2 + C2 (4-n)^2 + .... + Cn (2n-n)^2 = n.2^...

    Text Solution

    |

  9. Show that C0 + 3C1 + 5C2 + .... +(2n+1) Cn = (n+1)(2^n)

    Text Solution

    |

  10. Find the sum of the following C1 - 2C2 + 3C3 - ..... + n(-1)^(n-1) Cn

    Text Solution

    |

  11. Find the sum of the following 1.2 C(2) + 2.3 C(3) + ... + (n-1)nCn

    Text Solution

    |

  12. Find the sum of the following C1 + 2^2 C2 + 3^2 C3 + ... + n^2Cn

    Text Solution

    |

  13. Find the sum of the following C1 - 2C2 + 3C3 - ..... + n(-1)^(n-1) Cn

    Text Solution

    |

  14. Show that C1^2 + 2C2^2 + 3C3^2 + ... + "^nCn^2 = ((2n-1!))/{(n-1)!}^2

    Text Solution

    |

  15. Show that C2 + 2C3 + 3C4 + ... + (n-1)Cn = 1+ (n-2) 2^(n-1)

    Text Solution

    |

  16. prove that :-C1 - 1/2 C2 + 1/3 C3 + ... + (-1)^(n+1) 1/n Cn = 1+1/2 + ...

    Text Solution

    |

  17. C0 C1 + C1 C2 + .... + C(n-1) Cn = (2^n.n.1.3.5... (2n-1))/(n+1)

    Text Solution

    |

  18. The sum 1/(1!9!) + 1/(3!7!) + ... + 1/(7!3!) + 1/(9!1!) can be written...

    Text Solution

    |

  19. Using binomial theorem show that 1^(99) + 2^(99) +3^(99) + 4^(99) + 5^...

    Text Solution

    |

  20. Using the binomial theorem show that 1^(99) + 2^(99) +3^(99) + 4^(99) ...

    Text Solution

    |