SOLUTION: In this continued fraction, the numbers directly in front of the addition signs alternate 1, 2, 1, 2, ... infinitely after the first repeated 1. Find the value of the fraction.

Algebra ->  Problems-with-consecutive-odd-even-integers -> SOLUTION: In this continued fraction, the numbers directly in front of the addition signs alternate 1, 2, 1, 2, ... infinitely after the first repeated 1. Find the value of the fraction.       Log On


   

Question 1199949: In this continued fraction, the numbers directly in front of the
addition signs alternate 1, 2, 1, 2, ... infinitely after the first repeated 1. Find the value of the fraction.
Repeated Fraction Diagram: https://ibb.co/LRJcnLB
Thank you!
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: sqrt%283%29
This is the same as writing sqrt(3)

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Explanation:

Let x be equal to the continued fraction given to us.

This value is positive because each piece we're adding is positive.

Now subtract 1 from both sides

This is done so that the repeating portions (1,2,1,2,...) are isolated on the right hand side.
Focus on the left-most part of each row. Ignore the numerators of 1.

The stuff after "2+...", highlighted in the red box, can be replaced with x-1 because of the repeated nature of this sequence.
The x-1 represents the massive nested fraction block.



x+-+1+=+1%2F%281+%2B+1%2F%282%2Bhighlight%28%28x-1%29%29%29%29

x+-+1+=+1%2F%281+%2B+1%2F%282%2B%28x-1%29%29%29

x+-+1+=+1%2F%281+%2B+1%2F%28x%2B1%29%29

x+-+1+=+1%2F%28%28x%2B1%29%2F%28x%2B1%29+%2B+1%2F%28x%2B1%29%29

x+-+1+=+1%2F%28%28x%2B1%2B1%29%2F%28x%2B1%29%29

x+-+1+=+1%2F%28%28x%2B2%29%2F%28x%2B1%29%29

x+-+1+=+%28x%2B1%29%2F%28x%2B2%29

%28x-1%29%28x%2B2%29+=+x%2B1

x%5E2%2Bx-2+=+x%2B1

x%5E2%2Bx-x+=+1%2B2

x%5E2+=+1%2B2

x%5E2+=+3

x+=+%22%22%2B-+sqrt%283%29

x+=+sqrt%283%29 or x+=+-sqrt%283%29

Keep in mind that x cannot be negative since each of the infinitely many fractional pieces we're adding are positive.

Positive + positive = positive

This rules out x+=+-sqrt%283%29 and we're left with x+=+sqrt%283%29 as the final answer.

Therefore,


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You can use coding software to generate each convergent which will help partially verify the answer.
I have done so as indicated below:
1 + 1/1 = 2
1 + 1/(1+1/2) = 1.66666666666667
1 + 1/(1+1/(2 + 1/(1 + 1/2))) = 1.72727272727273
1 + 1/(1+1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/2))))) = 1.73170731707317
1 + 1/(1+1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/2))))))) = 1.73202614379085
sqrt(3) = 1.73205080756888

Admittedly the precision isn't too great, but we have 4 decimal digits matching.
I.e. we're able to get to 1.7320, then the rest of the decimal digits are incorrect.
This process is continued until reaching the desired level of precision.
Each decimal value is approximate.

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Another way to numerically approximate x is to compute the pieces in reverse order.
Rather than go left to right, I'm working right to left. Furthermore, I'm starting on the inside and working toward the outside of this massive nested fraction. This is a recursive process.
Let ans = 2
1 + 1/ans = 1.5
2 + 1/ans = 2.66666666666667
1 + 1/ans = 1.375
2 + 1/ans = 2.72727272727273
1 + 1/ans = 1.36666666666667
2 + 1/ans = 2.73170731707317
1 + 1/ans = 1.36607142857143
2 + 1/ans = 2.73202614379085
1 + 1/ans = 1.36602870813397
2 + 1/ans = 2.73204903677758
1 + 1/ans = 1.36602564102564
2 + 1/ans = 2.73205068043172
1 + 1/ans = 1.36602542081759

I used software to generate the list above.
For any given line, each 'ans' refers to the answer of the previous line.

The stopping criteria is anything starting with "1+1/ans" to match what was the denominator for the upper-most level of x-1.
If we stopped at 1.36602542081759, then the reciprocal of this is 1/1.36602542081759 = 0.73205079844083
Add on 1 to get 1.73205079844083 which is somewhat close to sqrt(3) = 1.73205080756888

Compare the digits
1.73205079844083 ...... computed from continued fraction
1.73205080756888 ...... calculator's display of sqrt(3)
The stuff in red is what matches.

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Further Reading:
https://sites.millersville.edu/bikenaga/number-theory/periodic-continued-fractions/periodic-continued-fractions.html
https://pi.math.cornell.edu/~gautam/ContinuedFractions.pdf
https://mathworld.wolfram.com/ContinuedFraction.html
https://en.wikipedia.org/wiki/Continued_fraction
https://www.planetmath.org/TableOfContinuedFractionsOfSqrtnFor1N102
https://brilliant.org/wiki/continued-fractions/