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

1 answer

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 \(\boxed{252}\).