Question

The sum of the first 21 terms of the progression -18-15-12.......is

Answer 1

To find the sum of the first 21 terms of the arithmetic progression (AP) given by \(-18, -15, -12, \ldots\), we need to identify the first term, the common difference, and then apply the formula for the sum of the first \(n\) terms of an AP.

1. **Identify the first term (\(a\)) and the common difference (\(d\))**:
- The first term \(a = -18\).
- The common difference \(d\) can be calculated as:
\[
d = -15 - (-18) = -15 + 18 = 3
\]

2. **Use the sum formula for the first \(n\) terms of an AP**:
The sum \(S_n\) of the first \(n\) terms of an arithmetic progression can be calculated using the formula:
\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\]
Where:
- \(n = 21\) (the number of terms)
- \(a = -18\) (the first term)
- \(d = 3\) (the common difference)

3. **Substitute the values into the formula**:
\[
S_{21} = \frac{21}{2} \times (2 \times -18 + (21-1) \times 3)
\]
Simplifying inside the parentheses:
\[
= \frac{21}{2} \times (2 \times -18 + 20 \times 3)
\]
\[
= \frac{21}{2} \times (-36 + 60)
\]
\[
= \frac{21}{2} \times 24
\]

4. **Calculate the sum**:
\[
S_{21} = \frac{21 \times 24}{2}
\]
\[
= \frac{504}{2} = 252
\]

Therefore, the sum of the first 21 terms of the progression is \(\boxed{252}\).