Question

Give a recursive definition of the set of positive integers that are multiples of 5.

Short answer

$\begin{array}{r}5\in S\\ s+5\in S\text{whenever}s\in S\end{array}$

Step-by-step solution

The recursive definition of the sequence:

The recursive sequence is a sequence of numbers indexed by an integer and generated by solving a recurrence equation.

 Step 2: To give a recursive definition of the set of positive integers that are multiples of 5.

It is given that S is the set of positive integers that are multiples of 5.

Now, the first positive integer that is a multiple of 5 is 5.

$5\in \mathrm{S}$

Hence, every multiple of 5 is the previous multiple of 5 increased by 5.

$\mathrm{s}+5\in \mathrm{S}$ whenever $\mathrm{s}\in \mathrm{S}$.