Posted by **Cody Libby** on Tuesday, August 30, 2011 at 1:10am.

I NEED to give the equation 5+3/(5+3/(5+3/(5+3/(..... as an exact answer. i know it involves using x and exponents...

- ALGEBRA PLEASE HELP -
**Anonymous**, Tuesday, August 30, 2011 at 1:53am
Please write exact formula.

- ALGEBRA PLEASE HELP -
**Reiny**, Tuesday, August 30, 2011 at 5:47am
I think I am looking at a repeating "continued fraction"

let x = 5+3/(5+3/(5+3/(5+3/(.....

now look at the part that I put in bold

x = 5+3/(**5+3/(5+3/(5+3/(..... **

Isn't the part in bold the same as the original x ?

(since it also goes to infinity, it would not matter at which 5 you start)

so we have

x = 5 + 3/x

x^2 = 5x + 3

x^2 - 5x - 3 = 0

x = (5 ± √(25 - 4(1)(-3))/2

= (5 + √37)/2 , rejecting the negative answer, since obviously the expression is positive, with only positive numbers showing up

so 5+3/(5+3/(5+3/(5+3/(..... = **(5+√37)/2**

check: my result is appr equal to 5.54139...

using : 5 + 3/5 = 5.6

using : 5 + 3/(5+3) = 5.375

using : 5 + 3/(5 + 3/5) = 5.53714..

using : 5 + 3/(5 + 3/(5 + 3)) = 5.55814..

notice that each consecutive result converges on my exact answer of

(5+√37)/2 , once above it, then below it, the difference getting smaller each time.

- ALGEBRA PLEASE HELP -
**Cody Libby**, Tuesday, August 30, 2011 at 1:24pm
thank you very much for your help. i was stuck on what value to put as x

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