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Interesting Observation of Progression of Perfect Squares

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Joined: 30 Apr 2008
Posts: 1867

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Location: Oklahoma City
Schools: Hard Knocks
Interesting Observation of Progression of Perfect Squares [#permalink]

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29 May 2009, 21:51
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<Randomness>

I'm not surew why I just now noticed this, but if you take the progression of perfect squares, you can find the next one by adding the next odd integer to the current perfect square, starting with adding 1 to 0 since 0 is technically the first perfect square.
0 (+1)
1 (+3)
4 (+5)
9 (+7)
16 (+9)
25 (+11)
36 (+13)
49 (+15)
64 (+17)
81 (+19)
100 (+21)
121

You see how the difference in the perfect squares is a pattern?

I have no idea what significance this has, but maybe someone out there with a deeper level of mathematics theory can shed some light on this, as well as explain whether this has any significance.

<End Randomness>
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J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a\$\$.

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Kudos [?]: 615 [2], given: 32

SVP
Joined: 07 Nov 2007
Posts: 1792

Kudos [?]: 1059 [0], given: 5

Location: New York
Re: Interesting Observation of Progression of Perfect Squares [#permalink]

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29 May 2009, 22:27
jallenmorris wrote:
<Randomness>

I'm not surew why I just now noticed this, but if you take the progression of perfect squares, you can find the next one by adding the next odd integer to the current perfect square, starting with adding 1 to 0 since 0 is technically the first perfect square.
0 (+1)
1 (+3)
4 (+5)
9 (+7)
16 (+9)
25 (+11)
36 (+13)
49 (+15)
64 (+17)
81 (+19)
100 (+21)
121

You see how the difference in the perfect squares is a pattern?

I have no idea what significance this has, but maybe someone out there with a deeper level of mathematics theory can shed some light on this, as well as explain whether this has any significance.

<End Randomness>

n , n+1 are consecutive integers

(n+1)^2 - n^2 = n^2 +1+2n -n^2 = 2n+1

So. For any perfect square n^2 to get the next perfect square, we need to add (2n+1).

n=10
n^2 =100
(n+1)^2 = 100 + (2n+1) = 100+(20+1) =121
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Kudos [?]: 1059 [0], given: 5

Re: Interesting Observation of Progression of Perfect Squares   [#permalink] 29 May 2009, 22:27
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