Sequences & Series
Sum of infinite geometric series
Grade 11
Question:
<p>Given \(\dfrac{\left(\dfrac{a}{b}\right)}{\left(\dfrac{b-1}{b}\right)} = 4\), find \(S = \dfrac{\left(\dfrac{a}{a+b}\right)}{\left(1 - \dfrac{1}{a+b}\right)} = \dfrac{a}{a+b-1}\).</p>
Step-by-Step Solution
Key Concept: Simplify the given equation by converting complex fractions to obtain a relationship between a and b, then substitute into the target expression. The key is recognizing that dividing by a fraction equals multiplying by its reciprocal.
<p><strong>Step 1:</strong> Simplify the given equation.</p><p>$$\frac{\frac{a}{b}}{\frac{b-1}{b}} = 4$$</p><p>Dividing by a fraction means multiplying by its reciprocal:</p><p>$$\frac{a}{b} \cdot \frac{b}{b-1} = 4$$</p><p>$$\frac{a}{b-1} = 4$$</p><p>Therefore: <strong>a = 4(b-1) = 4b - 4</strong></p><p><strong>Step 2:</strong> Simplify the target expression S.</p><p>$$S = \frac{\frac{a}{a+b}}{1 - \frac{1}{a+b}}$$</p><p>The denominator simplifies: $$1 - \frac{1}{a+b} = \frac{a+b-1}{a+b}$$</p><p>$$S = \frac{a}{a+b} \cdot \frac{a+b}{a+b-1} = \frac{a}{a+b-1}$$</p><p><strong>Step 3:</strong> Substitute a = 4b - 4.</p><p>$$S = \frac{4b-4}{(4b-4)+b-1} = \frac{4b-4}{5b-5} = \frac{4(b-1)}{5(b-1)} = \frac{4}{5}$$</p><p>∴ Answer: <strong>4/5</strong></p>
Correct Answer: 4/5