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# Rational Number

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107
1

This is a continued fraction of a rational number in the form of a/b: 3, 3, 3, 3, 3, 3, 3, 3, 2. Need some help in finding a/b. Any hint or help would be appreciated. Thank you.

Nov 23, 2018

### 1+0 Answers

#1
+101181
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3, 3, 3, 3, 3, 3, 3, 3, 2

We take this this in reverse order to find the continued fraction....it is actually not as hard as it seems...just a  (rather tedious) iterative process

So....We work from right to left

Take the last 2, invert it ⇒ 1/2....add it to the next-to-the-last 3   = 3 + 1/2 = 7/2

Invert this = 2/7 and add it to the next 3 to the left = 3 + 2/7 = 23/7

Invert this =  7/23 and add it to the next 3 to the left   = 3 + 7/23 = 76/23

Invert this and add it to the next  3 on the left = 23/76 + 3 = 251/76

Invert this and add it to the next 3 on the left = 76/251 + 3  = 829/251

Invert this and add it to the next 3 on the left = 251 / 829 + 3 = 2738 / 829

Invert this and add it to the next 3 on the left = 829 / 2738 + 3 = 9043/2738

Invert this and add it to the last 3 on the left = 2738/9043 + 3  = 29867 / 9043  =  a / b

Nov 23, 2018
edited by CPhill  Nov 23, 2018