Let 𝑧𝑛=𝑎𝑛+𝑖𝑏𝑛 be a series of complex numbers located on the unit circle.
given that 𝑎1+𝑎2+⋯+𝑎10=2.
Let 𝑤𝑛 be a series defined by 𝑤𝑛=𝑧𝑛+1𝑧𝑛.
Calculate the sum of the first ten terms of 𝑤𝑛 .

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.

Learn how to calculate the sum of series of complex numbers on the unit circle with a detailed step-by-step solution. Understand the concepts of complex numbers, unit circle, and telescoping series. Suitable for advanced high school students.

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