Question Find the sum of the first 34 terms of the following series, to the nearest integer. $$ 8,12,16, \ldots $$ Answer Attempt t out of 3 Submit Answer

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UpStudy AI · Accepted Answer

To find the sum of the first 34 terms of the series $$ 8, 12, 16, \ldots $$, we first identify the characteristics of the series.

1. **Identify the first term and common difference:**
   - The first term $$ a = 8 $$
   - The common difference $$ d = 12 - 8 = 4 $$

2. **Determine the number of terms:**
   - We need to find the sum of the first $$ n = 34 $$ terms.

3. **Use the formula for the sum of an arithmetic series:**
   The sum $$ S_n $$ of the first $$ n $$ terms of an arithmetic series can be calculated using the formula:
   $$
   S_n = \frac{n}{2} \times (2a + (n - 1)d)
   $$

4. **Substituting the known values into the formula:**
   - $$ n = 34 $$
   - $$ a = 8 $$
   - $$ d = 4 $$

Now, substituting these values into the formula:
   $$
   S_{34} = \frac{34}{2} \times (2 \times 8 + (34 - 1) \times 4)
   $$

5. **Calculating step by step:**
   - First, calculate $$ 2a $$:
     $$
     2a = 2 \times 8 = 16
     $$
   - Next, calculate $$ (n - 1)d $$:
     $$
     (34 - 1) \times 4 = 33 \times 4 = 132
     $$
   - Now, add these results:
     $$
     2a + (n - 1)d = 16 + 132 = 148
     $$
   - Finally, calculate $$ S_{34} $$:
     $$
     S_{34} = \frac{34}{2} \times 148 = 17 \times 148
     $$

6. **Perform the multiplication:**
   $$
   17 \times 148 = 2516
   $$

Thus, the sum of the first 34 terms of the series is $$ 2516 $$.

7. **Rounding to the nearest integer:**
   Since $$ 2516 $$ is already an integer, the final answer is:
$$
\boxed{2516}
$$
The sum of the first 34 terms is 2516.

Question Find the sum of the first 34 terms of the following series, to the nearest integer. \[ 8,12,16, \ldots \] Answer Attempt t out of 3 Submit Answer

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