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Thread: Quadratic equation proof

  1. #1
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    Quadratic equation proof

    Consider the equation $\displaystyle x^2-x-c=0, \ \ \ x>0 $
    Define the sequence $\displaystyle \{ x_n \} $ by $\displaystyle x_{n+1} = \sqrt {c+x_n} \ \ \ x_1 >0 $

    So that the sequence converges monotonically to the solution of the equation.

    Proof so far.

    I have $\displaystyle x_2 = \sqrt {c +x_1 } $ and $\displaystyle x_3 = \sqrt {c+x_2} = \sqrt {c + \sqrt {c+x_1 } } > \sqrt {c+x_1} = x_2$

    So this sequence is monotonically increasing, but do I need to do induction to be more convincing?

    I want to show that this sequence is bounded, so it does converge. Then suppose it converges to L. But I'm stuck here, guess I didn't start this right.

    Any hints? Thanks.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tttcomrader View Post
    Consider the equation $\displaystyle x^2-x-c=0, \ \ \ x>0 $
    Define the sequence $\displaystyle \{ x_n \} $ by $\displaystyle x_{n+1} = \sqrt {c+x_n} \ \ \ x_1 >0 $

    So that the sequence converges monotonically to the solution of the equation.

    Proof so far.

    I have $\displaystyle x_2 = \sqrt {c +x_1 } $ and $\displaystyle x_3 = \sqrt {c+x_2} = \sqrt {c + \sqrt {c+x_1 } } > \sqrt {c+x_1} = x_2$

    So this sequence is monotonically increasing, but do I need to do induction to be more convincing?
    yes, use induction. can you take it from there?

    I want to show that this sequence is bounded, so it does converge. Then suppose it converges to L. But I'm stuck here, guess I didn't start this right.

    Any hints? Thanks.
    yes, say the limit is L, then what do we know? $\displaystyle \lim x_{n + 1} = \lim x_n$. so that, from our equation we have, $\displaystyle L = \sqrt{c + L}$. now solve for L. this would complete the proof after you prove convergence
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  3. #3
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    I think I'm completely lost on this one.

    First, is $\displaystyle \sqrt {c+ \sqrt {c+x_1 } } > \sqrt {c+x_1} $? It really depends on what c is.

    Second, I have $\displaystyle L = \sqrt { c+L } $, then obtains $\displaystyle L^2 - L - c = 0 $, now how do I solve for L?

    Thanks.
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  4. #4
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    $\displaystyle L = \frac{{1 \pm \sqrt {1 + 4c} }}{2}$
    Be careful to check for domain problems.
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