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.