Miss Bennet has put an interpolating polynomial through your seven points,
she could have in fact added an eighth point with value, say 7176, and put
a polynomial throught these eight points.
She could also have added:
where is any function on to her polynomial and it would still interpolate
your data exactly.
Without some background information constraining solutions, such puzzles have no unique solution.
Just to clarify, the formula I provided above is on the assumption that the terms of your sequence satisfy a polynomial equation of degree at most 6.
In general, if the first k terms of a sequence are given, and it is assumed that the terms of the sequence satisfy a polynomial equation of degree at most , then the nth term of the sequence is
Alternatively, you could also let and solve for the coefficients by substituting the first given values – but this is going to take a hell of a much longer time.