Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>Consider an A.P. \(a_1, a_2, a_3, \ldots\) such that \(a_3 + a_5 + a_8 = 11\) and \(a_4 + a_2 = -2\), then the value of \(a_1 + a_6 + a_7\) is</p>
<p>\(-8\)</p>
<p>5</p>
<p>7</p>
<p>9</p>
Step-by-Step Solution
Key Concept: In an A.P., express all terms using first term (a) and common difference (d), then use the two given conditions to form simultaneous equations to find a and d, finally compute the required sum.
<p><strong>Step 1:</strong> Set up the A.P. with first term <em>a</em> and common difference <em>d</em>.</p><p>General term: a<sub>n</sub> = a + (n-1)d</p><p><strong>Step 2:</strong> Use condition 1: a₃ + a₅ + a₈ = 11</p><p>[a + 2d] + [a + 4d] + [a + 7d] = 11</p><p>3a + 13d = 11 ... (i)</p><p><strong>Step 3:</strong> Use condition 2: a₄ + a₂ = -2</p><p>[a + 3d] + [a + d] = -2</p><p>2a + 4d = -2</p><p>a + 2d = -1 ... (ii)</p><p><strong>Step 4:</strong> Solve equations (i) and (ii).</p><p>From (ii): a = -1 - 2d</p><p>Substitute in (i): 3(-1 - 2d) + 13d = 11</p><p>-3 - 6d + 13d = 11</p><p>7d = 14 ⟹ <strong>d = 2</strong></p><p>Therefore: a = -1 - 2(2) = <strong>-5</strong></p><p><strong>Step 5:</strong> Calculate a₁ + a₆ + a₇</p><p>a₁ = -5</p><p>a₆ = -5 + 5(2) = 5</p><p>a₇ = -5 + 6(2) = 7</p><p>a₁ + a₆ + a₇ = -5 + 5 + 7 = <strong>7</strong></p><p>∴ Answer: C</p>
Correct Answer: C