First, your sequence is: 1, 1, 5, 13, 25, 41, 61
Take the differences between adjacent terms:0, 4, 8, 12, 16, 20
Take the difference again: 4, 4, ,4 ,4, 4
Notice that all the second differences are the same. This means the polynomial will be second order so the sequence must look like:
a*n^2 + b*n + c
Now its a matter of solving for a, b, and c. If you call this last difference (here its 4) k and your highest exponent m (here that is 2) then a = k/m!
To get the next coefficient (b) subtract a*n^2 from each term so the terms become:
1 - 2*1^2 = -1
1 - 2*2^2 = -7
You don't need to do the whole sequence, you only need to do one term more than the exponent you are trying to solve for. you get -6, -6, -6, -6, etc.
Using the formula for coefficients above b = -6/1! = -6
So the sequence formula so far is 2*n^2 - 6*n + c
To find c, just evaluate for n=1: 2*1^2 - 6*1 + c = 1
So: c=5
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why is difference -6 not 6? why is formula for c equal to 1?
2007-12-18
15:55:46
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4 answers
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asked by
Anonymous
in
Mathematics