The recursively defined sequence a1=k an=(1)/(2)(an-1+(k)/(an-1)), if n>1 can be

step 1 Let's first put the rule into a calculator. I'm using decimal's calculator. And the second part

The recursively defined sequence a1=k ...

The recursively defined sequence $$ \begin{array}{l} a_{1}=k \\ a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right), \quad \text { if } n>1 \end{array} $$ can be used to compute $\sqrt{k}$ for any positive number $k$. This sequence was known to Sumerian mathematicians 4000 years ago, and it is still used today. Use this sequence to approximate the given square root by finding $a_{6} .$ Compare the result with the actual value. (Source: Heinz-Otto, P., Chaos and Fractals, Springer-Verlag.) (a) $\sqrt{2}$ (b) $\sqrt{11}$

The recursively defined sequence
$$
\begin{array}{l}
a_{1}=k \\
a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right), \quad \text { if } n>1
\end{array}
$$
can be used to compute $\sqrt{k}$ for any positive number $k$. This sequence was known to Sumerian mathematicians 4000 years ago, and it is still used today. Use this sequence to approximate the given square root by finding $a_{6} .$ Compare the result with the actual value. (Source: Heinz-Otto, P., Chaos and Fractals, Springer-Verlag.)
(a) $\sqrt{2}$
(b) $\sqrt{11}$

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Key Concepts

Recursive Sequences

A recursive sequence is one in which each term is defined in relation to one or more previous terms. Such sequences are essential in many areas of mathematics, as they allow complex behavior to be described by a simple rule, leading to analysis of patterns and convergence properties.

Babylonian Method (Heron’s Method)

The Babylonian method is an ancient algorithm for approximating square roots. It uses an iterative process where the current estimate is averaged with the quotient of the target number and the current estimate. This technique illustrates how iterative methods can efficiently converge to an accurate square root.

Iterative Approximation

Iterative approximation involves successively refining estimates to approach an accurate solution. In methods like the Babylonian approach, each iteration produces a more precise estimate of the square root, showcasing the power of repeated application of a simple calculation.

Convergence Analysis

Convergence analysis studies whether and how quickly a sequence or iterative method approaches a final value. In the context of square root approximation, it ensures that the sequence of estimates indeed gets progressively closer to the true square root, validating the method's effectiveness.

Transcript

00:01 Let's first put the rule into a calculator.

00:04 I'm using decimal's calculator.

00:05 And the second part of a a sub n is one half times a sub n minus 1.

00:15 So i'm just going to call that x for now plus k and k for the first one we're doing the spirit of 2.

00:22 So that would be 2 over a sub n minus 1 or x.

00:28 And then we're going to complete a table.

00:32 We don't want these in here yet.

00:35 All right, we'll come back to that in a moment.

00:45 So for the square root of two, if we want to go to a sub six.

00:58 So given that the first term is two in this case, then we want to know when x is two because that would get substituted for a sub one and the next one, and we get 1 .5.

01:13 And then the census is recursive, that's a sub 2, a sub 3 is by plugging 1 .5 in for x.

01:26 And then 1 .416 repeating.

01:37 Again, this is our first, second, third, fourth.

01:47 A sub 5 would be 1 .4142157.

01:53 An a sub six 1 .4142136 which we get the same output so this is approximately 1 .414 -2136 the actual value of the square root of 2 is 1 .414 -2 -1 -3 -6 so very very close now i want to do this again with 11.

02:43 So i am going to alter this slightly and that if the first term is 11, then we put 11 in for x into our second part...

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