Question

explain why a recursive rule alone does not define only one sequence

Answer 1

Suppose you have a sequence \(a_n=a_{n-1}\) for \(n\ge2\). This is clearly a constant sequence, since \(a_2=a_1\), \(a_3=a_2\), and so on. But what's \(a_1\)? We need a concrete value as our starting point in order to have a proper sequence.