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Define the function $p_n(t)=(-1)^n (-t)_n$.
1. Calculate $p_3(2)$.
Solution: Calculate
$$\begin{array}{ll}
p_3(2) &= (-1)^3 (-2)_3 \\
&= (-1) [(-2)(-1)(0)] \\
&= 0.
\end{array}$$
2. Calculate $p_2(3)$.
Solution: Calculate
$$\begin{array}{ll}
p_2(3) &= (-1)^2 (-3)_2 \\
&= 1[(-3)(-2)] \\
&= 6.
\end{array}$$
3. Calculate $p_4(3)$.
Solution: Calculate
$$\begin{array}{ll}
p_4(3) &= (-1)^4 (-3)_4 \\
&= 1[(-3)(-2)(-1)(0)] \\
&= 0.
\end{array}$$
4. Calculate $\Delta p_3(t)$. Express your answer as a number multiplied to some $p_m(t)$.
Solution: Calculate
$$\begin{array}{ll}
\Delta p_3(t) &= p_3(t+1)-p_3(t) \\
&= (-1)^3(-(t+1))_3 - (-1)^3 (-t)_3 \\
&= (-1)^3 \left[ (-t-1)(-t)(-t+1) - (-t)(-t+1)(-t+2) \right] \\
&= (-1)^3 \left[ (-t)(-t+1) \right] \left[ (-t-1) - (-t+2) \right] \\
&= -\left[ (-1)^2 (-t)_2 \right] \left[ -3 \right] \\
&= 3 p_2(t)
\end{array}$$
5. Calculate $\Delta p_4(t)$. Express your answer as a number multiplied to some $p_m(t)$.
Solution: Calculate
$$\begin{array}{ll}
\Delta p_4(t) &= p_4(t+1)-p_4(t) \\
&= (-1)^4 (-(t+1))_4 - (-1)^4 (-t)_4 \\
&= (-1)^4 \left[ (-t-1)(-t)(-t+1)(-t+2) - (-t)(-t+1)(-t+2)(-t+3) \right] \\
&= -\left[ (-1)^3 (-t)_3 \right] \left[ (-t-1) - (-t+3) \right] \\
&= - p_3(t) [-4] \\
&= 4p_3(t).
\end{array}$$
6. Do you notice a pattern developing for your answers to numbers 4 and 5? What is it?
Solution: For any $m=1,2,3,\ldots$,
$$\Delta p_m(t) = mp_{m-1}(t),$$
i.e. this is some kind of a "discrete power rule".
