posted by on .

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...

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.