1. ## perfect squares

How many integers between 200 and 300 are perfect squares?

2. We know 10 x 10 = 100.

Working from there:

11 x 11 = 121
12 x 12 = 144
13x 13 = 169
14 x 14 = 196

15x15 = 225 but that's over 200.
So there are four perfect squares between 100 and 200 (not inclusive of 100)

3. Originally Posted by sri340
How many integers between 200 and 300 are perfect squares?
Essentially, the same idea as Gusbob:

$\displaystyle \sqrt{300} = 17.32...$

$\displaystyle \sqrt{200} = 14.14...$

How many integers between those two numbers?

4. 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

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### how many perfect squares between 14 and 15

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