Home
Class 12
MATHS
The value of sum(r=1)^(n+1)(sum(k=1)^n^k...

The value of `sum_(r=1)^(n+1)(sum_(k=1)^n^k C_(r-1))(w h e r er ,k ,n in N)` is equal to `2^(n+1)-2` b. `2^(n+1)-1` c. `2^(n+1)` d. none of these

A

`2^(n+1)-2`

B

`2^(n+1)-1`

C

`2^(n+1)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(r=1)overset(n+1)sum(underset(k=1)overset(n)sum.^(k)C_(r-1))= underset(r=1)overset(n+1)sum(underset(k=1)overset(n)sum(.^(k+1)C_(r)-.^(k)C_(r)))`
`= underset(r=1)overset(n+1)sum(.^(n+1)C_(r)+.^(1)C_(r))`
`= 2^(n+1) - 2`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise (Multiple)|27 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise (Comprehension)|20 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.8|10 Videos

  • AREA UNDER CURVES

    CENGAGE|Exercise Question Bank|10 Videos

  • CIRCLE

    CENGAGE|Exercise MATRIX MATCH TYPE|6 Videos

Similar Questions

Explore conceptually related problems

The value of sum_(r=1)^(n+1)(sum_(k=1)^n "^k C_(r-1)) ( where r ,k ,n in N) is equal to a. 2^(n+1)-2 b. 2^(n+1)-1 c. 2^(n+1) d. none of these

The value of sum_(r=0)^(3n-1)(-1)^r 6nC_(2r+1)3^r is

The value of sum_(r=1)^(n)(-1)^(r-1)((r )/(r+1))*^(n)C_(r ) is

Find the sum of sum_(r=1)^n(r^n C_r)/(n C_(r-1) .

The value of int_0^1 ( pi_(r=1)^n (x+r)) (sum_(k=1)^n 1/(x+k)) dx

Find the sum sum_(r=1)^n r^2(^n C_r)/(^n C_(r-1)) .

The value of sum_(k=0)^(n)(i^(k)+i^(k+1)) , where i^(2)= -1 , is equal to

The value of lim_(n->oo)sum_(r=0)^(n) (sum_(t=0)^(r-1)1/(5^n)*"^n C_r * "^r C_t .(3^t)) is equal to

The value of lim_(n rarroo) sum_(r=1)^(n)(1)/(sin{((n+r)pi)/(4n)}).(pi)/(n) is equal to

CENGAGE-BINOMIAL THEOREM-Exercise (Single)
  1. The value of sum(r=0)^(10)(r)^(20)Cr is equal to 20(2^(18)+^(19)C(10))...

    Text Solution

    |

  2. [(^n C0+^n C3+)1//2(^n C1+^n C2+^n C4+^n C5]^2+3//4(^n C1-^n C2+^n C4-...

    Text Solution

    |

  3. The value of sum(r=1)^(n+1)(sum(k=1)^n^k C(r-1))(w h e r er ,k ,n in ...

    Text Solution

    |

  4. The sum sumsum(0leilejle10) (""^(10)C(j))(""^(j)C(r-1)) is equal to

    Text Solution

    |

  5. The value of "^(1000)C(50)+^(999)C(49)+^(998)C(48)+......+^(950)C(0) i...

    Text Solution

    |

  6. If sum(r=0)^n{ar(x-alpha+2)^r-br(alpha-x-1)^r}=0, then prove that bn-(...

    Text Solution

    |

  7. If sum(r=0)^(2n)ar(x-2)^r=sum(r=0)^(2n)br(x-3)^ra n dak=1 for all kgeq...

    Text Solution

    |

  8. The value of underset(r=0)overset(40)sumr.^(40)C(r).^(30)C(r) is

    Text Solution

    |

  9. If ^n+1C(r+1) : ^n Cr : ^(n-1)C(r-1)=11 :6:3, then n r =

    Text Solution

    |

  10. If the three consecutive in the expansion of (1+x)^n are 28, 56, and 7...

    Text Solution

    |

  11. Which term in the expansion of (2-3x)^(19) has algebrically the last c...

    Text Solution

    |

  12. The value of ("^n C0)/n+("^nC1)/(n+1)+("^nC2)/(n+2)+....+("^nC n)/(2n)...

    Text Solution

    |

  13. The value of sum(r=1)^n(-1)^(r+1)(^^n C r)/(r+1) is equal to -1/(n+1) ...

    Text Solution

    |

  14. If sum(r=0)^(10)((r+2)/(r+1))^^n Cr=(2^8-1)/6 , then n is 8 b. 4 c. 6 ...

    Text Solution

    |

  15. The value ofsum(r=0)^(10) (-1)^(r).4^(10-r)""^(30)C(r)""^(30-r)C(10-r)...

    Text Solution

    |

  16. The value ofunderset(r=0)overset(10)sumr^(10)C(r),3^(r).(-2)^(10-r) is...

    Text Solution

    |

  17. ""^(15)C(0)=

    Text Solution

    |

  18. If (1+x)^n=C0+C1x+C2x^2++Cn x^n ,t h e nC0C2+C1C3+C2C4++C(n-2)Cn= ((2...

    Text Solution

    |

  19. ^404C4-^303C4.^4C1+^202C4.^4C2-^101C4.^4C3=

    Text Solution

    |

  20. The value of sum(r=0)^(40) r""^(40)C(r)""^(30)C(r) is

    Text Solution

    |