Home
Class 11
MATHS
If a > b and n is a positive integer, th...

If `a > b and n` is a positive integer, then prove that `a^n-b^n > n(a b)^((n-1)//2)(a-b)dot`

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

If n ge 1 is a positive integer, then prove that 3^(n) ge 2^(n) + n . 6^((n - 1)/(2))

If n is an odd positive integer, then a^(n)+b^(n) is divisible by

If n is an even positive integer, then a^(n)+b^(n) is divisible by

Prove that 2^n > n for all positive integers n.

If a ,\ m ,\ n are positive integers, then {root(m)root (n)a}^(m n) is equal to (a) a^(m n) (b) a (c) a^(m/n) (d) 1

If a\ a n d\ b are distinct integers, prove that a^n-b^n is divisible by (a-b) where n in NN .

If a and b are positive, use mathematical induction to prove that ((a+b)/(2))^(n) le (a^(n)+b^(n))/(2) AA n in N

Prove that [(n+1)//2]^n >(n !)dot

If a and b are distinct integers, prove that a - b is a factor of a^n-b^n , whenever n is a positive integer.

If a > b >0, with the aid of Lagranges mean value theorem, prove that n b^(n-1)(a-b) < a^n -b^n < n a^(n-1)(a-b) , if n >1. n b^(n-1)(a-b) > a^n-b^n > n a^(n-1)(a-b) , if 0 < n < 1.

CENGAGE ENGLISH-SEQUENCES AND SERIES-All Questions
  1. Prove that a^4+b^4+c^4> a b c(a+b+c),w h e r ea ,b ,c > 0.

    Text Solution

    |

  2. Prove that the greatest value of x y is c^3/sqrt(2a b), if a^2x^4+b^2y...

    Text Solution

    |

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

    Text Solution

    |

  4. If y = sin^-1(10x) + pi/2 then find the value of dy/dx .

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |