Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>If the sum of \(n\) terms of an A.P. is given by \(S_n = a + bn + cn^2\), where \(a, b, c\) are independent of \(n\), then</p>
<p>\(a = 0\)</p>
<p>common difference of A.P. must be \(2b\)</p>
<p>common difference of A.P. must be \(2c\)</p>
<p>first term of A.P. is \(b + c\)</p>
Step-by-Step Solution
Key Concept: For an A.P., the sum formula Sₙ must have the form Sₙ = An² + Bn (no constant term), so comparing with S_n = a + bn + cn², we must have a = 0. The first term and common difference can be extracted from coefficients of n.
<p><strong>Step 1:</strong> For an A.P., the sum of n terms is Sₙ = An² + Bn, which is a quadratic with no constant term.</p><p><strong>Step 2:</strong> Comparing S_n = a + bn + cn² with standard form, we must have <strong>a = 0</strong> (constant term must vanish).</p><p><strong>Step 3:</strong> With a = 0: S_n = bn + cn²</p><p><strong>Step 4:</strong> First term: t₁ = S₁ = b + c</p><p><strong>Step 5:</strong> Second term: t₂ = S₂ - S₁ = 2b + 4c - (b + c) = b + 3c</p><p><strong>Step 6:</strong> Common difference: d = t₂ - t₁ = (b + 3c) - (b + c) = 2c</p><p><strong>Step 7:</strong> General term: tₙ = Sₙ - Sₙ₋₁ = [bn + cn²] - [b(n-1) + c(n-1)²] = b + c(2n - 1)</p><p><strong>Step 8:</strong> Verification: This is linear in n with first term (b+c) and common difference 2c ✓</p><p>∴ Answer: ACD (a = 0; specific relationships between first term, common difference, and coefficients b, c hold)</p>
Correct Answer: ACD