To find the sum of the first 20 terms of the series 8, 12, 16, ..., 96, we first need to identify the type of series we are dealing with. This series is an **arithmetic series** because the difference between consecutive terms is constant. ### Step 1: Identify the first term and the common difference - The **first term (a)** of the series is 8. - The **common difference (d)** can be calculated by subtracting the first term from the second term: \[ d = 12 - 8 = 4 \] ### Step 2: Determine the number of terms We are asked to find the sum of the first 20 terms. Thus, \( n = 20 \). ### Step 3: Find the last term To find the last term of the series, we can use the formula for the \( n \)-th term of an arithmetic series: \[ a_n = a + (n - 1) \cdot d \] Substituting the values we have: \[ a_{20} = 8 + (20 - 1) \cdot 4 \] Calculating this step-by-step: 1. Calculate \( 20 - 1 = 19 \). 2. Multiply by the common difference: \( 19 \cdot 4 = 76 \). 3. Add this to the first term: \( 8 + 76 = 84 \). So, the 20th term is 84, not 96. This indicates that the series does not actually reach 96 within the first 20 terms. ### Step 4: Calculate the sum of the first 20 terms The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic series is: \[ S_n = \frac{n}{2} \cdot (a + a_n) \] Substituting the values we have: - \( n = 20 \) - \( a = 8 \) - \( a_n = 84 \) Now, plug these values into the formula: \[ S_{20} = \frac{20}{2} \cdot (8 + 84) \] Calculating this step-by-step: 1. Calculate \( \frac{20}{2} = 10 \). 2. Calculate \( 8 + 84 = 92 \). 3. Multiply: \( 10 \cdot 92 = 920 \). Thus, the sum of the first 20 terms of the series is **920**. ### Step 5: Evaluate the options Now, let's look at the options provided: - A. 1400 - B. 1040 - C. 960 - D. 920 The correct answer is **D. 920**. ### Explanation of why other options are incorrect: - **A. 1400**: This value is too high and does not correspond to the calculated sum of the first 20 terms. - **B. 1040**: This value is also too high and does not match the sum we calculated. - **C. 960**: This value is close but still incorrect; it does not reflect the correct sum of the first 20 terms. ### Summary of Key Points: - The series is arithmetic with a first term of 8 and a common difference of 4. - The 20th term is 84, not 96. - The sum of the first 20 terms is calculated using the formula for the sum of an arithmetic series. - The correct answer is **920** (Option D). This thorough breakdown should help you understand how to approach similar problems in the future!