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Define the function pn(t)=(−1)n(−t)n.
1. Calculate p3(2).
2. Calculate p2(3).
3. Calculate p4(3).
Consider the following calculation of Δp2(t): since p2(t)=(−1)2(−t)2=(−t)2, we may calculate
Δp2(t)=(−(t+1))2−(−t)2=(−t−1)2−(−t)2=(−t−1)(−t)−(−t)(−t+1).
We may express this as a multiple of (−t)1 because
Δp2(t)=(−t−1)(−t)−(−t)(−t+1)=[(−t)][(−t−1)−(−t+1)]=(−t)1(−2)=2[(−1)1(−t)1]=2p1(t).
4. Calculate Δp3(t). Express your answer as a number multiplied to some pm(t).
5. Calculate Δp4(t). Express your answer as a number multiplied to some pm(t).
6. Do you notice a pattern developing for your answers to numbers 4 and 5? What is it?