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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?