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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...


    Please write exact formula.


    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.


    thank you very much for your help. i was stuck on what value to put as x

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