Home
Class 11
MATHS
If Cr=(n !)/([r !(n-r)]), the prove tha...

If `C_r=(n !)/([r !(n-r)]),` the prove that `sqrt(C_1)+sqrt(C_2)+.......sqrt(C_n) lt sqrt(n(2^n-1)) ="">

Text Solution

AI Generated Solution

Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise All Questions|9 Videos

  • STRAIGHT LINES

    CENGAGE ENGLISH|Exercise All Questions|491 Videos

Similar Questions

Explore conceptually related problems

lim_(n to oo)[(sqrt(n+1)+sqrt(n+2)+....+sqrt(2n))/(n sqrt((n)))]

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .

If n=12 m(m in N), prove that ^n C_0-(^n C_2)/((2+sqrt(3))^2)+(^n C_4)/((2+sqrt(3))^4)-(^n C_6)/((2+sqrt(3))^6)+= (-1)^m((2sqrt(2))/(1+sqrt(3)))^ndot

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

Prove that sum_(r=0)^(2n)(r. ^(2n)C_r)^2=n^(4n)C_(2n) .

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)) .

With the notation C_r= ^nC_2 = (n!)/(r!(n-r)! ) when n is positive inteer let S_n=C_n-(2/3)C_(n-1)+(2/3)^2C_(n-2)+- ……..+ (-1)^n(2/3)^n.C_0

Prove that .^(n)C_(r )+.^(n-1)C_(r )+..+.^(r )C_(r )=.^(n+1)C_(r+1)

Prove that sum_(r=0)^n r(n-r)(^nC_ r)^2=n^2(^(2n-2)C_n)dot

CENGAGE ENGLISH-SEQUENCES AND SERIES-All Questions
  1. If y = sin^-1(10x) + pi/2 then find the value of dy/dx .

    Text Solution

    |

  2. If a+b=1,a >0, prove that (a+1/a)^2+(b+1/b)^2geq(25)/2dot

    Text Solution

    |

  3. If Cr=(n !)/([r !(n-r)]), the prove that sqrt(C1)+sqrt(C2)+.......sq...

    Text Solution

    |

  4. If x and y are positive real numbers and m, n are any positive integ...

    Text Solution

    |

  5. The least value of the expression 2(log)(10)x-(log)x(0. 01),forx >1, i...

    Text Solution

    |

  6. If a , b , c , are positive real numbers, then prove that (2004, 4M) {...

    Text Solution

    |

  7. True / False For every intger n >1 , the inequality (n !)^(1//n)<(n+1)...

    Text Solution

    |

  8. If x ,y in R^+ satisfying x+y=3, then the maximum value of x^2y is.

    Text Solution

    |

  9. For anyx ,y , in R^+,x y >0 . Then the minimum value of (2x)/(y^3)+(x...

    Text Solution

    |

  10. If a ,b ,a n dc are positive and 9a+3b+c=90 , then the maximum value o...

    Text Solution

    |

  11. Given that x ,y ,z are positive real such that x y z=32. If the minimu...

    Text Solution

    |

  12. If the product of n positive numbers is n^n , then their sum is (a)...

    Text Solution

    |

  13. If a,b,c are different positive real numbers such that b+c−a,c+a−b and...

    Text Solution

    |

  14. Find the greatest value of x^2 y^3, where x and y lie in the first qua...

    Text Solution

    |

  15. Find the maximum value of (7−x)^4 (2+x)^5 when x lies between −2 and ...

    Text Solution

    |

  16. If a1, a2,...... ,an >0, then prove that (a1)/(a2)+(a2)/(a3)+(a3)/(a4...

    Text Solution

    |

  17. If a > b and n is a positive integer, then prove that a^n-b^n > n(a b)...

    Text Solution

    |

  18. If a ,b ,a n dc are positive and a+b+c=6, show that (a+1//b)^2+(b+1/...

    Text Solution

    |

  19. Prove that [(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)> x^x y^y z^z >[(x+y+z)/3]^(...

    Text Solution

    |

  20. Prove that 1^1xx2^2xx3^3xxxxn^nlt=[(2n+1)//3]^(n(n+1)//2),n in Ndot

    Text Solution

    |