Find the sum of the AP : -37. -33, -29, .... of first 12 terms

Question

Accepted Answer

To find the sum of the first 12 terms of the arithmetic progression (AP) given by the sequence -37, -33, -29, ..., we need to use the formula for the sum of the first $ n $ terms of an AP:

$$ S_n = \frac{n}{2} \times (2a + (n-1)d) $$

where:
- $ S_n $ is the sum of the first $ n $ terms,
- $ a $ is the first term,
- $ d $ is the common difference,
- $ n $ is the number of terms.

**Step-by-step Explanation:**

1. **Identify the first term ($ a $) and the common difference ($ d $):**
   - The first term $ a = -37 $.
   - The common difference $ d $ can be found by subtracting the first term from the second term: $ d = -33 - (-37) = 4 $.

2. **Determine the number of terms ($ n $):**
   - We are asked to find the sum of the first 12 terms, so $ n = 12 $.

3. **Apply the formula for the sum of the first $ n $ terms:**
   $$
   S_{12} = \frac{12}{2} \times (2 \times (-37) + (12-1) \times 4)
   $$
   $$
   S_{12} = 6 \times (-74 + 44)
   $$
   $$
   S_{12} = 6 \times (-30)
   $$
   $$
   S_{12} = -180
   $$

Thus, the sum of the first 12 terms of the AP is $-180$, which corresponds to Option 1.

**Explanation of Incorrect Options:**

- **Option 2: 180**
  - This option is incorrect because the calculation clearly shows that the sum is negative, not positive. The arithmetic progression is composed of negative numbers, and the sum of these terms results in a negative value.

- **Option 3: 810**
  - This option is incorrect because it is significantly larger than the actual sum. The calculation shows that the sum is $-180$, not a positive and large number like 810.

- **Option 4: -108**
  - This option is incorrect because it does not match the calculated sum. The correct calculation results in $-180$, not $-108$.

Therefore, the correct answer is indeed Option 1: $-180$.