How many integers between 200 and 300 are perfect squares?
You can also use a progression.
Say $\displaystyle T_n = n^2$, (note difference $\displaystyle D_n = (n + 1)^2 - n^2 = n^2 + 2n + 1 - n^2 = 2n + 1$).
How many terms between $\displaystyle 200$ and $\displaystyle 300$ ?
Start from the biggest square under $\displaystyle 200$, which is $\displaystyle 14^2 = 196$. Now many $\displaystyle D_n$ with $\displaystyle n > 14$ can you fit in $\displaystyle 300 - 196 = 104$ ?
Is there any sum of terms of $\displaystyle 2n + 1$ with $\displaystyle n > 14$ and starting from $\displaystyle 15$ consecutive that fit into $\displaystyle 104$ ? Sure ! We have $\displaystyle 31 + 33 + 35 < 104$, but we cannot fit $\displaystyle 37$, so there are only three perfect squares between $\displaystyle 200$ and $\displaystyle 300$.
There ... must be a better way to explain this, though ..
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I know this solution is a bit useless and boring after the previous answers, but eh, this is worth a post