4a. Any fixed real number may be approximated by a rational fraction with any de

Question

4a. Any fixed real number may be approximated by a rational fraction with any desired degree of accuracy if we allow the denominator b to become large enough. The size of the denominator is the price we pay for a good approximation. In the light of these observatons, how high is the price (i.e. the size of the denominator vs. the accuracy) which we pay as we approximate by the fractions These are called the convergents of the simple continued fraction for 4b. Describe (and justify, if possible) the process represented by the following table for calculating the suessive oonverge nts in Problem 4 of the continued fraction for Do you notice anything interesting about the 2x2 determinants? 1 11 2 2 3 4c. Use the process from Problem 5 to make a table for the fraction Ad. Use your results from Problems 5 and 6 to find an integral solution to each of the linear Diophanbne equabons: 29x + 11y = 1 and 7469x + 2463y = 1.

Explanation / Answer

Please specify the part u have problem in... As per QA guidelines I am solving the first part of the question:-

4(a)sol:- the series given in the question may be solved and rewritten as follows:-

2, 3/1, 8/3, 13/5,.....

As the series should approximately converge to 29/11

We have to find the nth term of term which approximately equals the value of 29/11 from the series

Now ,

Looking at terms in the numerator (leave out 2)

We find 3,5,8, .....,~29 are in Arithmetic progression with common difference 5

So by definition of AP nth term of Ap

29=3+5*(n-1)

Which implies n=6.12

Also the denominator of series formed by terms leaving 2,

1,3,5,.....~11 forms AP with common difference 2

So

11=1+2*(n-1)

Which implies n=6

Thus the 6th term of series (terms excluding 2)

Approximately equals 29/11

Putting n=6 in numerator series

3+5×5=28

Thus 28/11 appears in the series which approximately equals value of 29/11

For finding percentage accuracy,

Accuracy % = 100-error%

Error%= (29/11-28/11)÷29/11×100=3.448%

Thus accuracy%=100-3.448=96.551%

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