Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p><strong>(b)</strong> If the sum of the first \(2n\) terms of the A.P. \(2, 5, 8, \ldots\) is equal to the sum of the first \(n\) terms of the A.P. \(57, 59, 61, \ldots\), then \(n\) equals</p>
<p>(A) \(10\)</p>
<p>(B) \(12\)</p>
<p>(C) \(11\)</p>
<p>(D) \(13\)</p>
Step-by-Step Solution
Key Concept: Use the formula for the sum of n terms of an A.P.: S_n = n/2(2a + (n-1)d), then equate the sum of first 2n terms of the first A.P. with the sum of first n terms of the second A.P. to find n.
<p><strong>Step 1: Identify the A.P. parameters.</strong></p><p>For A.P. 2, 5, 8, ... : first term a₁ = 2, common difference d₁ = 3</p><p>For A.P. 57, 59, 61, ... : first term a₂ = 57, common difference d₂ = 2</p><p><strong>Step 2: Write the sum of first 2n terms of the first A.P.</strong></p><p>S₂ₙ = (2n)/2[2(2) + (2n-1)(3)]</p><p>S₂ₙ = n[4 + 6n - 3]</p><p>S₂ₙ = n[1 + 6n]</p><p>S₂ₙ = n + 6n²</p><p><strong>Step 3: Write the sum of first n terms of the second A.P.</strong></p><p>Sₙ = n/2[2(57) + (n-1)(2)]</p><p>Sₙ = n/2[114 + 2n - 2]</p><p>Sₙ = n/2[112 + 2n]</p><p>Sₙ = n[56 + n]</p><p>Sₙ = 56n + n²</p><p><strong>Step 4: Set the two sums equal.</strong></p><p>n + 6n² = 56n + n²</p><p>6n² - n² = 56n - n</p><p>5n² = 55n</p><p><strong>Step 5: Solve for n.</strong></p><p>5n² - 55n = 0</p><p>5n(n - 11) = 0</p><p>Since n ≠ 0, we have n = 11</p><p><strong>Step 6: Verify.</strong></p><p>S₂ₙ = 11 + 6(121) = 11 + 726 = 737</p><p>Sₙ = 56(11) + 121 = 616 + 121 = 737 ✓</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C