How many integers between 200 and 300 are perfect squares?

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- Dec 12th 2009, 02:51 PMsri340perfect squares
**How many integers between 200 and 300 are perfect squares?** - Dec 12th 2009, 03:25 PMGusbob
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) - Dec 13th 2009, 01:23 AMaidan
- Dec 13th 2009, 03:20 AMBacterius
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 :D