A polynomial f(x) has degree 8 and f(i)=2^i for i=0,1,2,3,4,5,6,7,8 . Find f(9).
Hmmm. I don't see any tricky way to get this. If f(x) has degree n and f(i) = 2^i for 0,...n then we have coefficients of
f(x) = ax^n + bx^(n-1) + ...
n coefficients
1: 1 1
2: 1/2 1/2 1
3: 1/6 0 5/6 1
4: 1/24 -1/12 11/24 7/12 1
I don't see any real pattern here.
I got these by using wolframalpha to solve matrix equations. The last one was the result of
solve {{0,0,0,0,1},{1,1,1,1,1},{16,8,4,2,1},{81,27,9,3,1},{256,64,16,4,1}}*{{a},{b},{c},{d},{e}} = {{1},{2},{4},{8},{16}}