Results 1 to 2 of 2

Math Help - mathematical induction - inequalities

  1. #1
    Newbie
    Joined
    Jun 2008
    Posts
    1

    mathematical induction - inequalities

    hi guys.. exams is in a few days need help on this topic!

    can i know how do i solve these questions?

    Q: Prove that the following inequalities hold for all n ∈ N.

    1. (1+x)^n\geq1+nx if x\geq-1

    2. 1^3+2^3+...+(n-1)^3 < \frac{1}{4}n^4 < 1^3+2^3+ ... +n^3

    and one more question on mathematical induction..

    Q: Prove by induction that the following statement is true for every positive integer n.

     x-y is a factor of x^n - y^n

    thanks in advance!!
    Last edited by zhin; June 16th 2008 at 08:30 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    Hi
    Quote Originally Posted by zhin View Post
    Q: Prove that the following inequalities hold for all n ∈ N.

    1. (1+x)^n\geq1+nx if x\geq-1
    • Prove that (1+x)^0\geq 1+0\times x (easy )
    • Assume that there exists en integer n such that (1+x)^n\geq1+nx (1). As we want to show that (1+x)^{n+1}\geq1+(n+1)x , you may try using (1+x)^{n+1}=(1+x)(1+x)^n. (expand this and apply (1))
    2. 1^3+2^3+...+(n-1)^3 < \frac{1}{4}n^4 < 1^3+2^3+ ... +n^3
    Notice that \frac{n^4}{4}=\int_0^nt^3\,\mathrm{d}t. As t\mapsto t^3 is an increasing function, \int_{k-1}^{k}(k-1)^3\,\mathrm{d}t<\int_{k-1}^{k}t^3\,\mathrm{d}t< \int_{k-1}^{k}k^3\,\mathrm{d}t which can be rewritten (k-1)^3<\int_{k-1}^{k}t^3\,\mathrm{d}t< k^3. Can you use this to show the two inequalities ?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Mathematical Induction - Inequalities
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: March 14th 2011, 08:44 AM
  2. Replies: 10
    Last Post: June 29th 2010, 01:10 PM
  3. Mathematical Induction [inequalities]
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: April 9th 2010, 08:35 AM
  4. Mathematical Induction to Prove Inequalities
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: November 28th 2009, 08:09 PM
  5. mathematical induction inequalities
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 27th 2007, 07:20 AM

Search Tags


/mathhelpforum @mathhelpforum