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Interesting Observation of Progression of Perfect Squares

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Interesting Observation of Progression of Perfect Squares [#permalink]

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

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

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New post 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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Interesting Observation of Progression of Perfect Squares

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