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July 14th, 2011, 03:39 PM   #1
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Arithmetic Progression

I have been trying to resolve this

= n/2(2*2+(n-1)*2)=n(n+1) and this is equal to 79*80


I got stuck when I got to 2n^2 +2n=12,640
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July 14th, 2011, 03:47 PM   #2
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By inspection, 2(n - 79)(n + 80).
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July 15th, 2011, 12:37 PM   #3
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Re: Arithmetic Progression

Kodwo,
From the looks of it, you knew (?) the following:
n*(n + 1) = 79*80
n*(n + 1) = 79*(79 + 1)

If that is the case, then there really is no easier case of "inspection". You can tell that if we replaced the "n"s on the left with "79", then there would be a true statement.

The approach that is less magical would be to solve the quadratic equation that you listed:
2n^2 + 2n - 12640 = 0
n^2 + n - 6320 = 0
Then you are looking for factors of 6320 that differ by 1. That in itself could be a chore for some, so you could resort to the quadratic formula.

In either case, you will get the provided solution.
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July 16th, 2011, 09:50 AM   #4
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A chore? The square root of 6320 is clearly slightly less than 80, so 79*80 would be the first thing to try.
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