1. ## Sequences and sums

Hi, I need help with a question regarding sequences and sums:

The upper of the two following sequences can be entered on the form of a polynomial (ie, an = degree 2 polynomial). Prove that the bottom can not be printed in the form of a degree 2 polynomial. (Hint: start by proving that the top can be entered on the form of a degree 2 polynomial).

0 1 4 9

0 1 4 10

Thanks!

2. Welcome to the forum.

It is not exactly clear what "can be entered on the form of a polynomial" means. Do you mean that there exists a polynomial $f(x)=ax^2+bx+c$ and four points $x_1,x_2,x_3,x_4$ such that $f(x_1)=0$, $f(x_2)=1$, $f(x_3)=4$, $f(x_4)=9$? Is there any condition on $x_i$, for example: $x_1=0$, ..., $x_4=3$ or $x_3=x_2+1$?

3. The quadratic function, f, such that f(0)= 0, f(1)= 1, f(2)= 4, f(3)= 9 is fairly obvious. If we are to assume that the second function must give f(0)= 0, f(1)= 1, f(2)= 4, f(3)= 10, then the answer is simple since a quadratic function, having 3 coefficients, is determined by 3 equations.