We are given that $z_n = a_n + i b_n$ is a series of complex numbers located on the unit circle, which implies that $|z_n| = 1$ for all $n$. The sum of the real parts $a_1 + a_2 + \cdots + a_{10} = 2$.

We need to calculate the sum of the first ten terms of the series $w_n$ defined by:

$w_n = z_{n+1} \overline{z_n}$

where $\overline{z_n}$ is the complex conjugate of $z_n$.

Step 1: Expressing $w_n$

The complex conjugate of $z_n$ is given by:

$\overline{z_n} = a_n - i b_n$

So,

$w_n = z_{n+1} \overline{z_n} = (a_{n+1} + i b_{n+1})(a_n - i b_n)$

Expanding the product:

$w_n = a_{n+1}a_n + b_{n+1}b_n + i(b_{n+1}a_n - a_{n+1}b_n)$

Since $|z_n| = 1$, $z_n$ can be written as $e^{i\theta_n}$, where $\theta_n$ is the argument of $z_n$. Therefore, $w_n$ can be written as:

$w_n = e^{i(\theta_{n+1} - \theta_n)}$

Step 2: Sum of the first ten terms of $w_n$

We need to calculate:

$\sum_{n=1}^{10} w_n = \sum_{n=1}^{10} e^{i(\theta_{n+1} - \theta_n)}$

This sum simplifies as a telescoping series:

$\sum_{n=1}^{10} w_n = e^{i(\theta_2 - \theta_1)} + e^{i(\theta_3 - \theta_2)} + \cdots + e^{i(\theta_{11} - \theta_{10})}$

Notice that:

$\sum_{n=1}^{10} w_n = e^{i(\theta_{11} - \theta_1)}$

Since $z_n$ is on the unit circle, $z_{11} = z_1$, and hence $\theta_{11} = \theta_1$. Therefore:

$e^{i(\theta_{11} - \theta_1)} = e^{i \cdot 0} = 1$

Thus, the sum of the first ten terms of $w_n$ is:

$\sum_{n=1}^{10} w_n = 1$

So, the sum of the first ten terms of $w_n$ is $\boxed{1}$.

Would you like more details on any of the steps or further exploration? Here are some related questions:

1. How does the concept of the unit circle affect the magnitude of complex numbers?
2. Can the sum $\sum_{n=1}^{10} w_n$ ever be different from 1 under different conditions?
3. How does the telescoping series simplify the sum in this problem?
4. What is the significance of the argument $\theta_n$ in complex numbers on the unit circle?
5. Can you generalize the result to any number $N$ instead of 10?

Tip: When dealing with complex numbers on the unit circle, expressing them in exponential form often simplifies calculations, especially for products and sums.