Problem:
Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.
Solution:
Let , and be the three terms, with . The given condition is
Consider the equation as a quadratic in and apply the quadratic formula to get
Because is an integer, the discriminant must be a perfect square. Hence
for some nonnegative integer . Because and have the same parity, they are both even. There are two ways to factor into two positive even integers. Thus either and or and , implying that or .
For , the sequence is .
For , the sequence is .
The requested sum is .
As in the previous solution, , from which .
Because is an integer, for some positive integer . Hence , which implies that or . Checking these two cases yields the same two possible values for as above.