Posted by Abigail on Wednesday, November 2, 2011 at 6:32pm.
These appear to be susceptible to solution using Newton's Divided Difference method.
For the first one,
n: 1 2 3 4 5
f(n): 6 28 64 114 178
d1(n)=f(n+1)-f(n): 22 36 50 64
d2(n)=[d1(n+1)-d1(n)]/2: 7 7 7 7
The second differences are constant. So, the final function has coefficients from the first entry in each row:
f(n) = 6 + 22(n-1) + 7(n-1)(n-2)
f(n) = 7x^2 + x - 2
This is the polynomial of lowest degree which fits all the given data points.
Read up on the method, and you can do the others easily. A web search on "divided difference interpolation formula" reveals lots of articles, and even a calculator at the ucla site.
pattern rule for
2,3, ___, 7, 11, 13, 17, ___
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