Sequences and sigma notation

The idea

A recurrence relation $u_{n+1} = f(u_n)$ builds a sequence one term at a time from a first term. A sequence is increasing, decreasing or periodic (it repeats). $\sum_{r=1}^{n}$ means add the terms from $r = 1$ to $r = n$.

1. Generate terms one at a time from $u_1$.
2. Look for the pattern: steady growth, or a cycle that repeats.
3. For a periodic sequence, use the remainder after dividing by the period.

Watch out Start from the right index: $u_1$ is the first term, so $u_2 = f(u_1)$.