Perfect Squares ending in 576 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 -
$$(1+3+5+7+\cdots+49) + \underbrace{(51+53+55+57+\cdots)}_{(n\text{ summands})}$$
Therefore, 
$$(1+3+5+7+\cdots+49) + \underbrace{(51+53+55+57+\cdots)}_{
(n \text{ summands})} = 1000k +576.$$ 
Since $(1+3+5+7+ ....49) = 576$, the equation reduces to
$$\underbrace{(51+53+55+57+\cdots)}_{n\text{ summands})} = 1000k$$
Using formula for Arithmetic Progression starting with 51 and a common difference of 2,
$$\begin{align*}
\frac{n}{2}\left(2(51) + (n-1)2\right) &= 1000k\\
n(n+50) &= 1000k
\end{align*}$$
Put $n = 100$, $100\times 150 = 1000k$, 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.

The flaw is

Since, (1+3+5+7+ ....49) = 576

The sum is actually 625. You need to stop a term earlier.
