To solve the problem, we need to express $z + \frac{1}{z}$ in the form $a + bi$, where $z = 1 - 2i$.

Step-by-Step Solution

1. Given:
   $z = 1 - 2i$.

2. Find $\frac{1}{z}$: To find the reciprocal of $z$, we multiply both numerator and denominator by the conjugate of $z$.
   The conjugate of $z = 1 - 2i$ is $1 + 2i$.

   $\frac{1}{z} = \frac{1}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i}$

   Simplify the denominator using the formula $(a - b)(a + b) = a^2 - b^2$:
   $(1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5$

   Therefore:
   $\frac{1}{z} = \frac{1 + 2i}{5} = \frac{1}{5} + \frac{2}{5}i$

3. Add $z + \frac{1}{z}$:
   Now add $z = 1 - 2i$ and $\frac{1}{z} = \frac{1}{5} + \frac{2}{5}i$:

   $z + \frac{1}{z} = (1 - 2i) + \left( \frac{1}{5} + \frac{2}{5}i \right)$

   Combine like terms (real and imaginary parts):

   • Real part: $1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5}$
   • Imaginary part: $-2i + \frac{2}{5}i = \left( -\frac{10}{5} + \frac{2}{5} \right)i = -\frac{8}{5}i$

   Thus:
   $z + \frac{1}{z} = \frac{6}{5} - \frac{8}{5}i$

4. Final Answer:
   The expression $z + \frac{1}{z}$ in the form $a + bi$ is:

   $\boxed{\frac{6}{5} - \frac{8}{5}i}$

Do you want me to clarify any part of the solution or show additional details? Let me know!

Related Questions:

1. How do you find the conjugate of a complex number?
2. What is the geometric representation of $z + \frac{1}{z}$ on the complex plane?
3. How do you multiply complex numbers in the form $a + bi$?
4. How do you compute the modulus of a complex number?
5. Can you explain why multiplying by the conjugate helps simplify $\frac{1}{z}$?

Tip: Always simplify complex expressions by separating real and imaginary parts clearly.