I want to find out perfect squares ending in 576, after the number 576.
Here is my derivation to arrive at such a number. Let the perfect square ending in 576 be 1000k+576. Every perfect square can be expressed as a the sum of a certain number of consecutive odd numbers. For eg: 2^2 = 1+3, 3^2 = 1+3+5, 4^2 = 1+3+5+7, and so on..
Hence I can write my required perfect square ending in 576 as -
(1+3+5+7+ ....49) + (51+53+55+57+....n terms)
Therefore, (1+3+5+7+ ....49) + (51+53+55+57+....n terms) = 1000k +576. Since, (1+3+5+7+ ....49) = 576, the equation reduces to
(51+53+55+57+....n terms) = 1000k
Using formula for Arithmetic Progression starting with 51 and a common difference of 2,
n/2[2(51) + (n-1)2] = 1000k
n(n+50) = 1000k
Put n = 100, 100*150 = 1000k, hence k = 15.
Put k = 15 in the perfect square term 1000k+576 we get the number 15576.
But 15576 is NOT a perfect square.
What is flawed in my derivation? Kindly help.
