Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>In an arithmetic progression whose first term is \(\alpha\) and common difference is \(\beta\); \(\alpha, \beta \neq 0\), the ratio \(r\) of the sum of the first \(n\) terms to the sum of \(n\) terms succeeding them, does not depend on \(n\). Then which of the following is/are correct?</p>
<p>\(\alpha : \beta = 2 : 1\)</p>
<p>If \(\alpha\) and \(\beta\) are roots of the equation \(ax^2 + bx + c = 0\) then \(2b^2 = 9ac\)</p>
<p>The sum of infinite G.P. \(1 + r + r^2 + \cdots\) is 3/2</p>
<p>If \(\alpha = 1\), then sum of 10 terms of A.P. is 100</p>
Step-by-Step Solution
Key Concept: For the ratio of sum of first n terms to sum of next n terms to be independent of n, the AP must satisfy a specific relationship between α and β. The sum of first n terms is S_n = n/2[2α + (n-1)β], and sum of next n terms is S_{2n} - S_n. Setting up the ratio and requiring it to be independent of n constrains α in terms of β.
<p><strong>Step 1:</strong> Set up the sum formulas. For an AP with first term α and common difference β:</p><p>S_n = n/2[2α + (n-1)β]</p><p>Sum of next n terms = S_{2n} - S_n = (2n)/2[2α + (2n-1)β] - n/2[2α + (n-1)β]</p><p>= n[2α + (2n-1)β] - n/2[2α + (n-1)β]</p><p>= n/2[2(2α + (2n-1)β) - (2α + (n-1)β)]</p><p>= n/2[4α + 4nβ - 2β - 2α - nβ + β] = n/2[2α + 3nβ - β]</p><p><strong>Step 2:</strong> Form the ratio r:</p><p>r = S_n / (S_{2n} - S_n) = [n/2(2α + (n-1)β)] / [n/2(2α + 3nβ - β)]</p><p>= [2α + nβ - β] / [2α + 3nβ - β]</p><p>= [2α - β + nβ] / [2α - β + 3nβ]</p><p><strong>Step 3:</strong> For r to be independent of n, the coefficients of n in numerator and denominator must have the same ratio as the constant terms:</p><p>β/(3β) = (2α - β)/(2α - β)</p><p>This gives: 1/3 = 1, which is impossible, OR the coefficient of n must be zero in both:</p><p>This requires: 2α - β = 0, hence <strong>α = β/2</strong></p><p>Then r = 0/0 form resolves to checking: if 2α = β, then r = nβ/(3nβ) = 1/3</p><p><strong>Step 4:</strong> Verify the constraint: The condition α = β/2 (or equivalently 2α = β) must hold.</p><p>∴ The condition is: <strong>2α - β = 0</strong>, and when satisfied, <strong>r = 1/3</strong> (constant)</p>
Correct Answer: ABCD