Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>Find the sum of the first 19 terms of the AP \(a_1, a_2, a_3, \ldots\) if it is known that \(a_1 + a_8 + a_{12} + a_{19} = 224\).</p>
Step-by-Step Solution
Key Concept: In an AP, terms equidistant from the ends have equal sums (a₁ + a₁₉ = a₈ + a₁₂). Use this symmetry property to find the sum of first and last terms, then apply the sum formula S_n = n/2(first + last).
<p><strong>Step 1:</strong> Write the general term. For AP with first term a₁ and common difference d:</p><p>a_n = a₁ + (n-1)d</p><p><strong>Step 2:</strong> Express each term in the given condition:</p><p>• a₁ = a₁</p><p>• a₈ = a₁ + 7d</p><p>• a₁₂ = a₁ + 11d</p><p>• a₁₉ = a₁ + 18d</p><p><strong>Step 3:</strong> Use the given condition a₁ + a₈ + a₁₂ + a₁₉ = 224:</p><p>a₁ + (a₁ + 7d) + (a₁ + 11d) + (a₁ + 18d) = 224</p><p>4a₁ + 36d = 224</p><p>a₁ + 9d = 56</p><p><strong>Step 4:</strong> Recognize that a₁ + 9d = a₁₀ (the middle term). Also note:</p><p>a₁ + a₁₉ = a₁ + (a₁ + 18d) = 2a₁ + 18d = 2(a₁ + 9d) = 2(56) = 112</p><p><strong>Step 5:</strong> Apply sum formula for 19 terms:</p><p>S₁₉ = (19/2)(a₁ + a₁₉) = (19/2)(112) = 19 × 56 = 1064</p><p><strong>∴ Answer: 1064</strong></p>
Correct Answer: 1064