recurrence relation question

Question

How can I build a recurrence equation if there isn't an $n$-variable? Example: $a_n = 3$. Also, how would I start making a recurrence equation for $a_n = 2n + 3$?

Answer 1

For the first one, you could use $a_0=3$ and $a_n=a_{n-1}$. For the second, put $a_0=3$ and $a_n=2+a_{n-1}$.

Ultimately, we need to determine what the relationship between subsequent terms is. In the first case, there is no difference at all (hence the $a_n=a_{n-1}$ bit). For the second, we note that each term will be $2$ more than the previous one (hence the $a_n=2+a_{n-1}$ bit). But that isn't enough. We also need to figure out what (at least) one $a_n$ is for these cases, or our recurrence relation suggests an infinite family of sequences, rather than a specific closed form. Fortunately, this part is easily done ("plug in" whatever $n$ you like).

What do you notice about the relationship between the subsequent terms of $a_n=3^n\cdot 5$? What about $a_n=1+n^2$? (The latter is tricker.)