Question

A sequence has constant 3rd differences of 24. What is the coefficient of the highest power term in its polynomial expression?

Answer 1

The constant third differences mean that the degree of the polynomial must be 3. If $a_0,a_1,a_2,$ are the terms of the sequence, then the third differences are \[\begin{array}{c@{~}c@{~}c@{~}c@{~}c@{~}c} &&&&&&a_0&&&&&&\\ &&&&&&\;\;\;\uparrow&&&&&&\\ &&a_1&&-8&&0&&-8&&a_0&&\\ &&\;\;\;\uparrow&&\;\;\;\uparrow&&\;\;\;\uparrow&&\;\;\;\uparrow&&\\ a_2&&-8&&-8&&-8&&-8&&a_0&&-8&&-8&&-8&&a_0 \end{array}\]Since the third differences are constant and there are 4 numbers in the diagonal, we have \[8+4d+6d+4d=24,\]which gives $d=2.$ Thus, the terms of the sequence are \[\begin{array}{c@{~}c@{~}c@{~}c@{~}c@{~}c} &&&&&&a_0&&&&&&\\ &&&&&&\;\;\;\uparrow&&&&&&\\ &&a_1&&2&&0&&2&&a_0&&\\ &&\;\;\;\uparrow&&\;\;\;\uparrow&&\;\;\;\uparrow&&\;\;\;\uparrow&&\\ a_2&&2&&2&&2&&2&&a_0&&2&&2&&2&&a_0 \end{array}\]and checking that this works, we find that $a_n = 2\binom{n}{2}a_0$ works. Thus, the sequence can be expressed as a polynomial: \[2\binom{n}{2}a_0 = a_0\cdot 2\cdot \frac{n(n-1)}{2} = a_0n(n-1).\]Thus, the coefficient of the highest power term is $\boxed{-1}.$