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We are given the complex number \( z = \frac{1 + i}{\sqrt{2}} \) and asked to compute the sum:

\[
1 + z + z^2 + z^3 + \cdots + z^{20}
\]

This is a geometric series with the first term as \( 1 \) and the common ratio as \( z \). To find the sum of this series, we can use the formula for the sum of a finite geometric series:

\[
S_n = \frac{1 - z^n}{1 - z}
\]

where \( n = 21 \) in our case (since we are summing terms from \( z^0 \) to \( z^{20} \)).

### Step 1: Understanding \( z \)
We have \( z = \frac{1 + i}{\sqrt{2}} \). This is a normalized complex number, where:

\[
z = e^{i \frac{\pi}{4}}
\]

This is because \( 1 + i \) in polar form has a modulus of \( \sqrt{2} \), and dividing by \( \sqrt{2} \) normalizes it, while its argument is \( \frac{\pi}{4} \).

### Step 2: Using the geometric sum formula
The sum formula becomes:

\[
S_{21} = \frac{1 - z^{21}}{1 - z}
\]

We now need to compute \( z^{21} \). Since \( z = e^{i \frac{\pi}{4}} \), using De Moivre's Theorem:

\[
z^{21} = e^{i \frac{21\pi}{4}}
\]

This can be simplified by reducing the argument modulo \( 2\pi \):

\[
\frac{21\pi}{4} = 5\pi + \frac{\pi}{4}
\]

which is equivalent to \( e^{i \frac{\pi}{4}} = z \). Thus, \( z^{21} = z \).

### Step 3: Substituting into the sum formula
Since \( z^{21} = z \), the sum becomes:

\[
S_{21} = \frac{1 - z}{1 - z}
\]

which simplifies to:

\[
S_{21} = 1
\]

Thus, the value of the sum \( 1 + z + z^2 + \cdots + z^{20} \) is **1**.

Would you like further clarification or details on any part of the solution? Here are some related questions to explore:

1. What is the general formula for the sum of a geometric series and when does it apply?
2. How can we express complex numbers in polar form?
3. What is De Moivre's theorem and how is it applied to powers of complex numbers?
4. Why is the modulus of the complex number \( z \) equal to 1?
5. What is the significance of reducing angles modulo \( 2\pi \) in complex exponentials?

**Tip:** When working with powers of complex numbers, converting to polar form makes calculations of high powers simpler using angles and modulus properties.

If z = (1 + i)/√2, what is the sum of the series 1 + z + z^2 + z^3 + ... + z^20?

Explore how to solve a geometric series where z = (1 + i)/√2, using the sum of a geometric series and De Moivre's Theorem. This solution involves expressing complex numbers in exponential form, reducing angles, and applying polar coordinates.

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