Sequence tests have no unique answer

Text

A test asks: “what is the next number in the sequence $2,\,4,\,6,\,8,\ ?$”. Prove that, for any chosen value $a$, there exists a polynomial $p$ with $p(1)=2$, $p(2)=4$, $p(3)=6$, $p(4)=8$ and $p(5)=a$. Conclude that the question has no mathematically forced answer.

Solution

The polynomial $q(x)=2x$ already fits the first four data. Add a term that vanishes at $x=1,2,3,4$: $p(x)=2x+c\,(x-1)(x-2)(x-3)(x-4).$ For $x=1,2,3,4$ the second summand is $0$, so $p$ still gives $2,4,6,8$. For $x=5$: $p(5)=10+c\,(4\cdot 3\cdot 2\cdot 1)=10+24c.$ Choosing $c=\dfrac{a-10}{24}$ gives $p(5)=a$. Since $a$ is arbitrary, every continuation is justifiable: the first four terms do not determine the fifth. $\blacksquare$