Find the sum of first 24 terms of the list of numbers whose nth term is a_n = 3 + 2n :

Question

Accepted Answer

To find the sum of the first 24 terms of the sequence where the nth term is given by $ a_n = 3 + 2n $, we can use the formula for the sum of an arithmetic sequence. The formula for the sum $ S_n $ of the first $ n $ terms of an arithmetic sequence is:

$$ S_n = \frac{n}{2} \times (a_1 + a_n) $$

where $ a_1 $ is the first term and $ a_n $ is the nth term.

**Step-by-step Explanation:**

1. **Identify the first term ($ a_1 $):**
   - For $ n = 1 $, the first term $ a_1 = 3 + 2(1) = 5 $.

2. **Identify the 24th term ($ a_{24} $):**
   - For $ n = 24 $, the 24th term $ a_{24} = 3 + 2(24) = 3 + 48 = 51 $.

3. **Calculate the sum of the first 24 terms ($ S_{24} $):**
   - Using the sum formula:
   $$
   S_{24} = \frac{24}{2} \times (5 + 51) = 12 \times 56 = 672
   $$

Thus, the sum of the first 24 terms is 672, which corresponds to Option 3.

**Explanation of why other options are incorrect:**

- **Option 1 (680):** This option is incorrect because the calculation of the sum using the arithmetic sequence formula results in 672, not 680. The arithmetic sequence formula has been applied correctly, and 680 does not match the calculated sum.

- **Option 2 (670):** This option is incorrect for the same reason as above. The correct sum, as calculated, is 672. The value 670 does not match the result obtained from the formula.

- **Option 4 (675):** This option is also incorrect. The calculated sum using the arithmetic sequence formula is 672, and 675 does not match this result.

In conclusion, the correct answer is Option 3 (672), as it accurately reflects the sum of the first 24 terms of the given sequence.