The squares of three positive integers are in arithmetic progression, and the third integer is 12 greater than the first. Find the second integer.
Let the 3 positive integers be x, y and z
but z = x+12
we have:
y^2 - x^2 = (x+12)^2 - y^2
2y^2 = 2x^2 + 24x + 144
y^2 = x^2 + 12x + 72
y = +√(x^2 + 12x + 72) , so we are looking for perfect squares for x^2 + 12x + 72
let x = 1, y = √85 , not an integer
let x = 2, y = √100 , well that was lucky
so x = 2, y = 10 and z = 14
check:
their squares are : 4, 100, and 196, which is in AP
is there any more info to help us answwer
Nope, there is no additional information.
oh okay hold on
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Why did the square go to the arithmetic progression party?
Because it wanted to hang out with the cool squares!
In this case, let's call the first positive integer "x". Since the squares are in arithmetic progression, the second positive integer would be "x + k" and the third positive integer would be "x + 2k" (where "k" is the common difference).
Now, we know that the third integer is 12 greater than the first.
So, we have the equation:
(x + 2k)^2 = x^2 + 12
Expanding and simplifying:
x^2 + 4kx + 4k^2 = x^2 + 12
Subtracting x^2 from both sides:
4kx + 4k^2 = 12
Dividing both sides by 4:
kx + k^2 = 3
Since k is a positive integer, the only possible values for k and x that satisfy this equation are k = 2 and x = 1.
Therefore, the second positive integer would be x + k = 1 + 2 = 3.
So, the second integer is 3.
To solve this problem, let's denote the three positive integers as a, b, and c. We are given that the squares of these integers, denoted as a^2, b^2, and c^2, are in arithmetic progression.
An arithmetic progression means that the difference between consecutive terms is constant. In this case, we have:
b^2 - a^2 = c^2 - b^2
Expanding both sides of the equation, we get:
b^2 - a^2 = c^2 - b^2
b^2 - a^2 = (c + b)(c - b)
Since we know that the third integer, c, is 12 greater than the first integer, a, we can write:
c = a + 12
Substituting this into the equation above, we have:
b^2 - a^2 = (a + 12 + b)(a + 12 - b)
b^2 - a^2 = (2a + 12)(12)
Expanding the right side of the equation, we get:
b^2 - a^2 = 24a + 144
Rearranging the equation, we have:
b^2 = 24a + 144 + a^2
We can now solve for the second integer, b, by considering different values of a and checking for valid solutions.
Let's try some values of a:
- If we set a = 1, then we have b^2 = 24 + 144 + 1^2 = 169. The square root of 169 is 13, so b = 13.
- If we set a = 2, then we have b^2 = 2^2 = 4, which doesn't satisfy the given condition of positive integers.
- If we set a = 3, then we have b^2 = 24 + 144 + 3^2 = 169, which again leads to b = 13.
Hence, the second integer, b, is 13.