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Define the function $p_n(t)=(-1)^n (-t)_n$.
1. Calculate $p_3(2)$.
2. Calculate $p_2(3)$.
3. Calculate $p_4(3)$.
Consider the following calculation of $\Delta p_2(t)$: since $p_2(t)=(-1)^2 (-t)_2 = (-t)_2$, we may calculate
$$\begin{array}{ll}
\Delta p_2(t) &= (-(t+1))_2 - (-t)_2 \\
&= (-t-1)_2 - (-t)_2 \\
&= (-t-1)(-t) - (-t)(-t+1).
\end{array}$$
We may express this as a multiple of $(-t)_1$ because
$$\begin{array}{ll}
\Delta p_2(t) &= (-t-1)(-t) - (-t)(-t+1) \\
&= [(-t)] [(-t-1) - (-t+1)] \\
&= (-t)_1 (-2) \\
&= 2 [(-1)^1 (-t)_1] \\
&= 2 p_1(t).
\end{array}$$
4. Calculate $\Delta p_3(t)$. Express your answer as a number multiplied to some $p_m(t)$.
5. Calculate $\Delta p_4(t)$. Express your answer as a number multiplied to some $p_m(t)$.
6. Do you notice a pattern developing for your answers to numbers 4 and 5? What is it?
