Find the sum of the first 22 terms of the AP : 8, 3, –2, . . .
Step-by-Step Solution
Key Concept: Use the sum formula for an arithmetic progression: \(S_n = \frac{n}{2}[2a+(n-1)d]\) or \(S_n = \frac{n}{2}(a+l)\), where \(a\) is the first term, \(d\) the common difference, \(n\) the number of terms and \(l\) the \(n^{th}\) term.
1. Identify the first term \(a\) and the common difference \(d\).
\[a = 8\]
\[d = 3-8 = -5\]
2. Number of terms required \(n = 22\).
3. Find the \(22^{nd}\) term (last term) \(l\) using \(l = a + (n-1)d\).
\[l = 8 + (22-1)(-5) = 8 + 21(-5) = 8 - 105 = -97\]
4. Apply the sum formula \(S_n = \frac{n}{2}(a + l)\).
\[S_{22} = \frac{22}{2}\bigl(8 + (-97)\bigr) = 11 \times (-89)\]
5. Calculate the product.
\[S_{22} = -979\]
Hence, the sum of the first 22 terms of the given AP is \(-979\).
Correct Answer: -979