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
To find the exact representation of the equation 5 + 3/(5 + 3/(5 + 3/(5 + 3/(...))), we can use the concept of a recurring decimal and solve it using algebraic manipulation.
Let's set x equal to the given expression:
x = 5 + 3/x
To solve for x, we need to isolate the x term. Rearranging the equation, we get:
x² = 5x + 3
Next, let's bring all terms to one side of the equation:
x² - 5x - 3 = 0
Now, we have a quadratic equation. We can solve it by using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation x² - 5x - 3 = 0, the values of a, b, and c are:
a = 1
b = -5
c = -3
Calculating the values, we find:
x = (-(-5) ± √((-5)² - 4(1)(-3))) / (2(1))
x = (5 ± √(25 + 12)) / 2
x = (5 ± √37) / 2
Thus, the exact solution to the equation 5 + 3/(5 + 3/(5 + 3/(5 + 3/(...))) is x = (5 ± √37) / 2.