How to prove that the recursive definition is correct?

Question

The recursive definition of the set of positive integers that are exponentiations of $3$ is:

1) $b_0=1$

2) $b_n=3(b_{n-1})$ for $n\ge 1$

Is it correct?

Now how can I prove that this definition is correct?

Answer 1

I assume you want to prove $$ b_n = 3^n \quad (n \in \mathbb{N}) $$ Induction seems natural to prove this.

But let us try something more fun. There is also a solution from the theory of linear recurrences:

The characteristic polynomial of the linear homogeneous recurrence with constant coefficients $$ b_n = 3 b_{n-1} $$ is $$ p(t) = t - 3 $$ with root $r=3$ and general solution $$ b_n = k \, 3^n $$ We derive the constant $k$ from the initial condition $$ b_0 = 1 $$ thus $$ 1 = k \, 3^0 $$ which gives $k=1$ and the solution $b_n = 3^n$.

Answer 2

You may try Mathematical induction to show that $b_n = 3^n$

For $n=0,$ the statement is $b_0 = 3^0 =1 $ which is true.

If the statement if true for $n$ , then $$ b_{n+1} = 3 b_n = 3(3^n)=3^{n+1}$$

Thus the statement is true for all $n\ge 0$