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.
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Identify the first term (\(a\)) and the common difference (\(d\)):
- The first term \(a = -18\).
- The common difference \(d\) can be calculated as: \[ d = -15 - (-18) = -15 + 18 = 3 \]
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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)
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Substitute the values into the formula: \[ S_{21} = \frac{21}{2} \times (2 \times -18 + (21-1) \times 3) \] Simplifying inside the parentheses: \[ = \frac{21}{2} \times (2 \times -18 + 20 \times 3) \] \[ = \frac{21}{2} \times (-36 + 60) \] \[ = \frac{21}{2} \times 24 \]
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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 \(\boxed{252}\).
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