If a sequence of numbers is determined by the equality and the values and , prove that:

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- Jun 9th 2009, 07:52 AMfardeen_genProve result for limits?
If a sequence of numbers is determined by the equality and the values and , prove that:

- Jun 9th 2009, 08:27 AMchisigma
Since is the mean between and is...

(1)

... so that is...

(2)

Kind regards

- Jun 9th 2009, 08:36 AMfardeen_gen
Could you explain the second line please? I couldn't understand it.

- Jun 9th 2009, 09:36 AMMedia_ManRewrite
Rewrite , for and . By substituting into the recursive formula, you get . Since we are taking iterative midpoints, the gap between successive values of are decreasing by half, so [tex]|a_n-a_{n-1}|=\left(\frac12\right)^{n-1}[/tex] . It should also be apparent that when n is odd, and when n is even, again, by virtue of iterating the midpoint formula, making the value of the gap alternate between positive and negative. Therefore, for , the limit of which as n gets large is .

- Jun 9th 2009, 10:01 AMpankaj
There is no need for such heavy explanation.

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