February 2026
First compute \(a_0^{(1)}\) and \(a_1^{(1)}\), and use this to compute \(a_2\). Can you then compute \(a_3\)?
Start by computing \(a_0, a_1,a_2,a_3,a_4\), and \(a_5\). Use these to compute the first few terms of the next layer (i.e., the terms \(a_0^{(1)}, a_1^{(2)}, \ldots\)). Then compute the first few terms of the next layer, and then the next layer. Keep going until you think you’ve reached the layer at which the sequence stabilises.
Try some specific examples of sequences like this. You already did one in Question 2. Try \(a_n = n^2 + 3n + 2\) and \(a_n = n^3 - n\). Explicitly compute expressions for the next few layers and pay attention to the degree of the polynomial that emerges at each layer.
Use Question 3 to guess at a general form for \(a_n\).