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:

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

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

How many integers between those two numbers?

4. You can also use a progression.
Say $T_n = n^2$, (note difference $D_n = (n + 1)^2 - n^2 = n^2 + 2n + 1 - n^2 = 2n + 1$).
How many terms between $200$ and $300$ ?

Start from the biggest square under $200$, which is $14^2 = 196$. Now many $D_n$ with $n > 14$ can you fit in $300 - 196 = 104$ ?

Is there any sum of terms of $2n + 1$ with $n > 14$ and starting from $15$ consecutive that fit into $104$ ? Sure ! We have $31 + 33 + 35 < 104$, but we cannot fit $37$, so there are only three perfect squares between $200$ and $300$.

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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