<p>Let \(a + ar_1 + ar_1^2 + \cdots + \infty\) and \(a + ar_2 + ar_2^2 + \cdots + \infty\) be two infinite series of positive numbers with the same first term. The sum of the first series is \(r_1\) and the sum of the second series is \(r_2\). Then the value of \((r_1 + r_2)\) is ___.</p>
Step-by-Step Solution
Key Concept: For a convergent geometric series with first term a and common ratio r, sum = a/(1-r). Set up equations using the given conditions that sum₁ = r₁ and sum₂ = r₂, then solve the resulting system algebraically.
<p><strong>Step 1:</strong> For the first series with first term <em>a</em> and common ratio <em>r₁</em>:</p><p>Sum = a/(1 - r₁) = r₁</p><p>∴ a = r₁(1 - r₁) = r₁ - r₁² ... (i)</p><p><strong>Step 2:</strong> For the second series with first term <em>a</em> and common ratio <em>r₂</em>:</p><p>Sum = a/(1 - r₂) = r₂</p><p>∴ a = r₂(1 - r₂) = r₂ - r₂² ... (ii)</p><p><strong>Step 3:</strong> Since both series have the same first term, equations (i) and (ii) give:</p><p>r₁ - r₁² = r₂ - r₂²</p><p>r₁ - r₂ = r₁² - r₂²</p><p>r₁ - r₂ = (r₁ - r₂)(r₁ + r₂)</p><p><strong>Step 4:</strong> Since r₁ ≠ r₂ (different series), divide both sides by (r₁ - r₂):</p><p>1 = r₁ + r₂</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1