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Mathematical Induction Problem

  1. Nov 3, 2005 #1
    Hi, all.
    I'm working on some proof by induction problems. While I understand the concept, this one threw me for a loop.
    Let [tex]x_1=\sqrt{2}[/tex] and [tex]x_{n+1}=\sqrt{2+x_n}[/tex]
    Show that [tex]x_n < x_{n+1}[/tex]
    I'd greatly appreciate help with this.
  2. jcsd
  3. Nov 3, 2005 #2
    sure you need to use induction? i would show that the stuff under the radical for [tex] x_{n+1}> x_n[/tex] we know this because [tex]x_n>0[/tex]. and 2 plus some other positive number will always be greater than two, and therefore the sq rt of that sum will be greater eh?
  4. Nov 3, 2005 #3


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    Uh, you certainly need induction. If xn = 98, then xn+1 = (2 + 98)1/2 = 1001/2 = 10 < 98 = xn.

    Show that x1 < x2
    Assume that xk < xk+1
    Use this to prove that xk+1 < xk+2
    Write out xk+1 and xk+2 in terms of xk. Then there xk+1 < xk+2 will follow immediately from xk < xk+1 as long as you know that the function f defined by f(a) = a1/2 is an increasing function.
  5. Nov 4, 2005 #4
    thanks...much clearer now

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