We are given the complex numbers $z_1 = 1 + 3i$ and $z_2 = 2 - i$, and we are asked to express $\frac{z_1 + z_2}{z_1 z_2}$ in the form $a + bi$, where $a$ and $b$ are real numbers.

Step 1: Calculate $z_1 + z_2$

$z_1 + z_2 = (1 + 3i) + (2 - i) = 3 + 2i$

Step 2: Calculate $z_1 z_2$

We use the distributive property to multiply $z_1$ and $z_2$:

$z_1 z_2 = (1 + 3i)(2 - i)$ $= 1 \cdot 2 + 1 \cdot (-i) + 3i \cdot 2 + 3i \cdot (-i)$ $= 2 - i + 6i - 3i^2$

Since $i^2 = -1$, this becomes:

$2 - i + 6i + 3 = 5 + 5i$

Step 3: Divide $z_1 + z_2$ by $z_1 z_2$

We now compute $\frac{3 + 2i}{5 + 5i}$. To do this, multiply the numerator and denominator by the conjugate of the denominator:

$\frac{3 + 2i}{5 + 5i} \times \frac{5 - 5i}{5 - 5i} = \frac{(3 + 2i)(5 - 5i)}{(5 + 5i)(5 - 5i)}$

First, calculate the denominator:

$(5 + 5i)(5 - 5i) = 25 - (5i)^2 = 25 - (-25) = 50$

Now, calculate the numerator:

$(3 + 2i)(5 - 5i) = 3 \cdot 5 + 3 \cdot (-5i) + 2i \cdot 5 + 2i \cdot (-5i)$ $= 15 - 15i + 10i - 10i^2$ $= 15 - 15i + 10i + 10 = 25 - 5i$

Thus, we have:

$\frac{25 - 5i}{50} = \frac{25}{50} - \frac{5i}{50} = \frac{1}{2} - \frac{1}{10}i$

Final Answer:

The expression in the form $a + bi$ is $\frac{1}{2} - \frac{1}{10}i$. This matches option (c).

Would you like further details on this solution?

Here are five related questions:

1. How do you multiply two complex numbers?
2. What is the geometric interpretation of multiplying complex numbers?
3. How do you find the conjugate of a complex number and why is it useful?
4. How do you divide complex numbers in polar form?
5. How can you express a complex number in polar and rectangular forms?

Tip: Always simplify complex expressions step by step, especially when multiplying or dividing by a conjugate!