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Riley Yates

Riley Yates

Answered question

2022-05-28

Let [ 1 ; 2 , 1 ¯ be a continued fraction. Find the corresponding number.

Answer & Explanation

antantil0

antantil0

Beginner2022-05-29Added 9 answers

For something like this, it becomes
x 1 = 1 2 + 1 x x 1 = 1 2 x + 1 x = x 2 x + 1 ,
Then, you can use the regular technique. If this is x, then
x 1 = 1 2 + 1 x x 1 = 1 2 x + 1 x = x 2 x + 1 ,
then you can solve.
In general, for b 1 , b 2 , , b n ¯ , instead of the denominators outside of the infinite fractions being continues b as suggested by b ¯ , it keeps cycling between b 1 , b 2 , b 3 , , b n .
Brooke Kramer

Brooke Kramer

Beginner2022-05-30Added 4 answers

Let
A = 1 + 1 2 + 1 1 + 1 2 + 1
Then notice that
A = 1 + 1 2 + 1 A
This gives
( A 1 ) ( 2 A + 1 ) = A
You can solve this quadratic.

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