Let x=sqrt(4+sqrt(4-sqrt(4+sqrt(4-...to infinity.Then x equals?

Question

Let x=sqrt(4+sqrt(4-sqrt(4+sqrt(4-...to infinity.Then x equals
(a) 3
(b) (sqrt(13)-1)/2
(c) (sqrt(13)+1)/2
(d) sqrt(13)

Answer 1

x = sqrt(4 + sqrt(4 - sqrt(4 + sqrt(4 - sqrt.....
x = sqrt(4 + sqrt(4 - x)
x² = 4 + sqrt(4 - x)
sqrt(4 - x) = x² - 4
4 - x = x^4 - 8x² + 16
x^4 - 8x² + x + 12 = 0

You can try to solve for this quartic using the method found in the site below.
(note: this equation is simpler because it is already in the depressed form.)

After getting the 4 solutions, try to use your common sense (or your calculator) to see that it converges to approx. 2.3027756....

^_^

Answer 2

Taki was on the right track, but made an arithmetic error. The correct answer is [C] (sqrt(13) + 1)/2. The solution goes as follows:

x = sqrt(4 + sqrt(4-sqrt(4+sqrt(4-sqrt(4+….))))
x^2 = 4 + sqrt(4-sqrt(4+sqrt(4-sqrt(4+….))))
x^2 – 4 = sqrt(4-sqrt(4+sqrt(4-sqrt(4+….))))
(x^2 – 4)^2 = 4 - sqrt(4+sqrt(4-sqrt(4+….))))
(x^2 – 4)^2 = 4 – x

x^4 – 8x^2 +x + 12 = 0

This equation has 4 real roots (I'll admit to resorting to a symbolic algebra program to find these roots):

(1 + sqrt(13))/2 ~= 2.303
(1 - sqrt(13))/2 ~=-1.303
(sqrt(17) - 1)/2 ~= 1.562
(-sqrt(17) - 1)/2 ~= -2.562

Now, the square root operation on a real positive number has two values, a positive and a negative. Assuming that one always takes the positive root in the original expression, we know that the value of this expression must be greater than the square root of 4 (i.e. x > 2) because x = sqrt(4 + epsilon), where epsilon is > 0. Therefore, the first solution to the 4th order equation is the solution we seek.

Answer 3

You do understand that the following image shows what you are essentially asking, right?

http://i2.photobucket.com/albums/y16/zorro1267/sqrt.gif

So this essentially comes down to solving:

x^2 -4 = Sqrt (4-x)

= x^4 - 8x^2 + x +12 = 0


Hmmm, maybe I had better reconsider that. One can form an infinite number of equations depending on how far one goes into the nesting. While one can say x^2 - 4 = Sqrt (4-x) one can go deeper into the nesting in the same manner and get a different polynomial formula. Perhaps this diverges. I don't feel like thinking about it.

Answer 4

3.

The equation equates out to Sqrt(4) +1

Answer 5

x = sqrt(4+sqrt(4+sqrt(4 +..)))
x ^ 2 = 4 + sqrt(4+sqrt(4 + sqrt(4 + ...)))
x ^ 2 = 4 + x
x^2 -x -4 = 0

=(-(-1)+sqrt((-1)^2 - 4(1)(-4)))/2(1)
= (1 + sqrt(17))/2
=2.56

=(-(-1)-sqrt((-1)^2 - 4(1)(-4)))/2(1)
= (1 - sqrt(1 + 16))/2
= (1 - sqrt(17))/2
=-1.56

Answer: 2.56 and -1.56 (None of the above)

Answer 6

a) 3

Answer 7

I've got to agree with Taki in that the correct answer is not represented. I cannot see any other way to do this other than a recursive formula. Writing a simple program to estimate it, and after a million recursions, I get the following

x~=2.561553

It has been many years since I've done any programming, so I don't exactly recall how to get more decimal places in the answer, however, as quickly as this converges, and after a million steps, I very highly doubt any significant change is going to come.


edit: oooops. missed the alternating signs. in that case, doing the same estimation yields that the answer is (c). I get that

x~= 2.30277563773

Answer 8

None of the above is correct.the limit will be oscillating without stop between two irrational values 1.302775638 and 2.302775638. if you don't believe me try it your self.

Answer 9

are you seriouse ?
get out of here , you are reminding me of my math teacher
he was a BEG A22HOLE