Math CentralQuandaries & Queries


Question from Sarah, a student:

The sum of the squares of two consecutive odd positive integers is 74. What are the integers?

Hi Sarah,

I think the easiest way to do this is just to try some consecutive, odd positive integers.

12 + 32 = 10
32 + 52 = 34
52 + 72 = 74

However I expect you are to use algebra. So how do you express two consecutive odd positive integers algebraically? An odd number is one more than an even number so if n is any non-negative integer 2n + 1 is odd. The next number is 2n + 2 which is eve and the next number is 2n + 3 which is odd. Thus 2n + 1 and 2n + 3 are two consecutive odd positive integers. Hence you need to find n if

(2n + 1)2 + (2n + 3)2 = 74

Expand this expression to obtain a quadratic in n. Factor the quadratic and solve for n. You already know that 2n + 1 = 5 and 2n + 3 = 7 so that's a check on your work.


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