In this problem, we are told that the ages of a family of four are in an arithmetic sequence, meaning the difference between consecutive ages is constant.

We know the following facts:

• The father is 45 years old.
• The youngest child is 5 years old.
• The middle child is 13 years older than the youngest child, meaning the middle child is $5 + 13 = 18$ years old.

Let’s define the ages of the family members as follows:

• Youngest child: $5$ years
• Middle child: $18$ years
• Father: $45$ years
• Mother's age: $x$ (which we are to find)

Since the ages are in an arithmetic sequence, the common difference between consecutive ages is constant. To find the common difference, observe that the difference between the youngest child and the middle child is $18 - 5 = 13$.

Thus, the difference between consecutive ages in this sequence is $13$. Therefore, the mother's age must be $18 + 13 = 31$ years old, but that does not match the answer choices.

It seems the arithmetic sequence might be arranged differently. Let’s check:

1. Youngest child: $5$ years
2. Middle child: $18$ years
3. Mother: $x$ years
4. Father: $45$ years

The common difference is $d$. So the mother's age should be: $18 + d = x$ and the father's age is: $x + d = 45.$

From this, we can solve for $d$ and $x$. Subtracting the two equations: $(45 - 18) = 2d \quad \Rightarrow \quad 27 = 2d \quad \Rightarrow \quad d = 13.5.$

Now, substitute $d = 13.5$ into $x = 18 + d$: $x = 18 + 13.5 = 31.5.$

This result isn't matching the options directly, let me revise quickly! .