Let's analyze the problem step-by-step.

Problem Analysis:

The equation $z^9 = 1$ describes the ninth roots of unity. These roots are complex numbers given by: $z_k = e^{2\pi i k / 9}, \quad k = 0, 1, 2, \ldots, 8$ where $i$ is the imaginary unit.

We are tasked with finding the value of $S$, which is defined as: $S = z^1 + z^4 + z^7 + z^1 + z^4 + z^7 + z^1$

Notice that $z^1, z^4, z^7$ repeat several times in the summation.

Simplification:

1. Group similar terms: Since $z^1, z^4,$ and $z^7$ are repeated, we simplify the sum: $S = 3z^1 + 3z^4 + 3z^7$

2. Factorize common terms: $S = 3(z^1 + z^4 + z^7)$

Thus, the problem reduces to finding $z^1 + z^4 + z^7$.

Property of Roots of Unity:

The ninth roots of unity are symmetrically distributed on the unit circle in the complex plane. A key property is: $\text{The sum of all ninth roots of unity is } 0.$ This is because: $\sum_{k=0}^{8} z_k = 1 + z^1 + z^2 + \cdots + z^8 = 0.$

Selection of Roots:

We are interested only in $z^1, z^4, z^7$. Observing the symmetry of roots, these three form an equilateral triangle on the unit circle. Their sum is: $z^1 + z^4 + z^7 = 0$

Final Calculation:

Substitute $z^1 + z^4 + z^7 = 0$ into $S$: $S = 3 \cdot 0 = 0$

Final Answer:

$S = 0$

Let me know if you would like further clarification or a deeper exploration of this concept! Here are five related questions for your learning:

1. What are the general properties of the $n$-th roots of unity?
2. How do the roots of unity relate to symmetry in the complex plane?
3. What happens when $n$ is a prime number in the equation $z^n = 1$?
4. How can roots of unity be used in Fourier transforms and signal processing?
5. How can you verify the symmetry of roots algebraically using trigonometric identities?

Tip:

When dealing with roots of unity, always consider their geometric interpretation on the unit circle to simplify summations.