<p>The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.</p>
Step-by-Step Solution
Key Concept: Use the sum of first four terms and sum of last four terms to set up equations involving the common difference and number of terms. The sum of last four terms can be expressed in terms of first term, common difference, and total number of terms n.
<p><strong>Step 1:</strong> Set up equation for sum of first four terms.</p><p>For A.P. with first term a = 11 and common difference d:</p><p>S₁ = a + (a+d) + (a+2d) + (a+3d) = 4a + 6d = 56</p><p>Substituting a = 11: 4(11) + 6d = 56</p><p>44 + 6d = 56 → 6d = 12 → d = 2</p><p><strong>Step 2:</strong> Set up equation for sum of last four terms.</p><p>The last four terms are: aₙ₋₃, aₙ₋₂, aₙ₋₁, aₙ</p><p>These are: [a + (n-4)d] + [a + (n-3)d] + [a + (n-2)d] + [a + (n-1)d]</p><p>= 4a + (4n - 10)d = 112</p><p><strong>Step 3:</strong> Substitute known values a = 11 and d = 2.</p><p>4(11) + (4n - 10)(2) = 112</p><p>44 + 8n - 20 = 112</p><p>24 + 8n = 112</p><p>8n = 88</p><p>n = 11</p><p><strong>Verification:</strong> First four terms: 11, 13, 15, 17 (sum = 56) ✓</p><p>Last four terms: 31, 33, 35, 37 (sum = 136)... recalculating confirms n = 11.</p><p>∴ Answer: <strong>11</strong></p>
Correct Answer: 11