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The formula provided in the image is related to the sum of an arithmetic series. There are two equivalent forms given for the sum of the first \(n\) terms of an arithmetic series:

1. \( S_n = \frac{n}{2} \left(2u_1 + (n-1)d \right) \)
2. \( S_n = \frac{n}{2} \left( u_1 + u_n \right) \)

Where:
- \( S_n \) is the sum of the first \( n \) terms.
- \( u_1 \) is the first term of the series.
- \( u_n \) is the \( n \)-th term of the series.
- \( d \) is the common difference between consecutive terms.
- \( n \) is the number of terms in the series.

### Applying this Formula to the Problem

The series provided in your previous question was \( \sum_{r=1}^{16} (5r - 3) \). This is an arithmetic series where:

- The first term \( u_1 \) can be found by substituting \( r = 1 \) into the expression \( 5r - 3 \):
  \[
  u_1 = 5(1) - 3 = 2
  \]
- The common difference \( d \) is the coefficient of \( r \) in the expression:
  \[
  d = 5
  \]
- The number of terms \( n = 16 \).
- The last term \( u_n \) can be found by substituting \( r = 16 \) into the expression:
  \[
  u_n = 5(16) - 3 = 80 - 3 = 77
  \]

Using the second formula for the sum of the series:

\[
S_{16} = \frac{16}{2} \left( u_1 + u_n \right) = \frac{16}{2} \left( 2 + 77 \right) = 8 \times 79 = 632
\]

### Final Answer:
\[
S_{16} = 632
\]

This confirms the earlier calculation.

Do you need any further clarification or details?

Here are 5 related questions for further practice:
1. Find the sum of the first 20 terms of the arithmetic series where the first term is 3 and the common difference is 4.
2. Calculate the sum of the first 10 terms of an arithmetic sequence where \( u_1 = 7 \) and \( u_{10} = 46 \).
3. Determine the 15th term of an arithmetic sequence where the first term is 5 and the common difference is 6.
4. Use the formula to find the sum of the first 25 terms of an arithmetic sequence with \( u_1 = -2 \) and \( d = 3 \).
5. Evaluate the sum of the first 12 terms of an arithmetic sequence where the common difference is 2 and the last term is 30.

**Tip:** Familiarity with both forms of the sum formula allows flexibility in solving problems, especially when different sets of information are provided.

Learn how to calculate the sum of an arithmetic series using the formula S_n = n/2 * (u_1 + u_n). Practice solving problems with detailed step-by-step solutions, covering arithmetic series concepts and formulas. Suitable for high school students.

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