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# Iteration

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The sequence of value given by the iterative formula is
X n+1=2/9-lnXn with initial value X=1,converges to alpha.

1) State an equation satisfied by alpha, and hence show that alpha is the x coordinate of a point on the curve where y=3.

2) Use this iterative formula to find alpha correct to 2 decimal places, showing each iteration.

##### Solution Summary

This provides a sequence given by an iterative formula, and shows how to find an equation satisfied by the convergence alpha and use the formula to find alpha.

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Solution: Assume that this sequence of value given by the iterative formula x(n+1)=2/9-ln[x(n)] with initial value x=1, converges to alpha, using letter 'a' to represent alpha. ie., x(n)->a as n->infinity.
Then, according to the iterative formula x(n+1)=2/9-ln[x(n)], we let n approach infinity, we have
a=2/9-ln(a) (1)
So the limit of this sequence satisfy equation (1).
From (1) we can rewrite it as ...

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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