It can be shown that for any positive integer n, the infinitely nested radical expression
(n+(n+(n+....)^1/2)^1/2)^1/2
equals a finite number. What is the largest positive integer n≤999 such that this expression is equal to a positive integer?
Details and assumptions
A nested radical expression is one which contains a radical inside another one, as in
((5^1/2)+3)^1/2
An infinitely nested radical expression is one in which the radicals continue to an infinite extent.
SO WHAT'S the Correct Answer ??
991
991 us wrong !!!
992
To solve this problem, we will start by simplifying the expression step by step until we can find a pattern.
Let's denote the given expression as E:
E = (n + (n + (n + ... )^1/2)^1/2)^1/2
Notice the pattern of how the expression is nested. We can rewrite E as:
E = (n + E^1/2)^1/2
Now, we will first solve for E:
E = (n + E^1/2)^1/2
To remove the square root, we can square both sides of the equation:
E^2 = n + E^1/2
Rearranging the equation, we get:
E^2 - E^1/2 - n = 0
This is a quadratic equation in terms of E. We can solve it using the quadratic formula:
E = (-(-1/2) ± √((-1/2)^2 - 4 * 1 * (-n))) / 2 * 1
Simplifying the expression inside the square root:
E = (1/2 ± √(1/4 + 4n)) / 2
Now, let's analyze the discriminant (1/4 + 4n) to determine the conditions for E to be a real positive integer.
For E to be a real positive integer, the discriminant must be a perfect square.
1/4 + 4n = m^2, where m is an integer
Multiplying both sides by 4:
1 + 16n = 4m^2
Rearranging the equation:
4m^2 - 16n = 1
Dividing both sides by 4:
m^2 - 4n = 1/4
This is a special case of the Pell's equation. The solutions to this equation can be found using continued fractions. However, since we are interested in finding the largest positive integer n ≤ 999, we can use trial and error to find the appropriate values.
Let's start by trying different values of n and solving for m:
For n = 0:
m^2 - 4(0) = 1/4
m^2 = 1/4
m = ±1/2, but we are looking for positive integers, so m = 1/2 is not valid.
For n = 1:
m^2 - 4(1) = 1/4
m^2 = 17/4
m ≈ ±2.062, which is not an integer.
For n = 2:
m^2 - 4(2) = 1/4
m^2 = 33/4
m ≈ ±2.561, which is not an integer.
We continue trying values of n until we find a positive integer solution for m.
For n = 3:
m^2 - 4(3) = 1/4
m^2 = 49/4
m ≈ ±3.561, which is not an integer.
For n = 4:
m^2 - 4(4) = 1/4
m^2 = 65/4
m ≈ ±4.062, which is not an integer.
For n = 5:
m^2 - 4(5) = 1/4
m^2 = 81/4
m ≈ ±4.561, which is not an integer.
For n = 6:
m^2 - 4(6) = 1/4
m^2 = 97/4
m ≈ ±5.062, which is not an integer.
We can see that there are no positive integer solutions for m when n ≤ 6.
Continuing this process, we find that for n = 7, we have a solution:
m^2 - 4(7) = 1/4
m^2 = 113/4
m ≈ ±5.561, which is not an integer.
For n = 8:
m^2 - 4(8) = 1/4
m^2 = 129/4
m ≈ ±6.062, which is not an integer.
For n = 9:
m^2 - 4(9) = 1/4
m^2 = 145/4
m ≈ ±6.561, which is not an integer.
Finally, for n = 10, we have:
m^2 - 4(10) = 1/4
m^2 = 161/4
m ≈ ±7.062, which is not an integer.
We can now conclude that the largest positive integer n ≤ 999 such that the expression E is equal to a positive integer is n = 7.
Therefore, the answer is n = 7.