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  1. #1
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    perfect squares

    How many integers between 200 and 300 are perfect squares?
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  2. #2
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    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)
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  3. #3
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    Quote Originally Posted by sri340 View Post
    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?
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  4. #4
    Super Member Bacterius's Avatar
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    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 ..

    -------------------------

    I know this solution is a bit useless and boring after the previous answers, but eh, this is worth a post
    Last edited by Bacterius; December 13th 2009 at 03:32 AM.
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