Sequences & Series
GP sum formula
Grade 11
Question:
<p>If \( (1-r)(1-2x-4x^2-8x^3-16x^4-32x^5) = 1 - r^6 \), \( (r \neq 1) \), then a value of \( \dfrac{r}{x} \) is</p>
<p>\( \dfrac{1}{2} \)</p>
<p>2</p>
<p>\( \dfrac{1}{4} \)</p>
<p>4</p>
Step-by-Step Solution
Key Concept: Recognize that the left side contains a geometric series in the bracket that can be summed using the formula for finite geometric series, allowing comparison with the right side.
<p><strong>Step 1:</strong> Rewrite the left side expression. Notice that:<br/>2x + 4x² + 8x³ + 16x⁴ + 32x⁵ = 2x(1 + 2x + 4x² + 8x³ + 16x⁴)</p><p><strong>Step 2:</strong> Recognize this as a geometric series with first term a = 1 and common ratio q = 2x. The sum is:<br/>1 + 2x + 4x² + 8x³ + 16x⁴ = (1-(2x)⁵)/(1-2x) = (1-32x⁵)/(1-2x)</p><p><strong>Step 3:</strong> Therefore:<br/>2x + 4x² + 8x³ + 16x⁴ + 32x⁵ = 2x · (1-32x⁵)/(1-2x)</p><p><strong>Step 4:</strong> Rewrite the original equation:<br/>(1-r)[1 - 2x - 4x² - 8x³ - 16x⁴ - 32x⁵] = 1 - r⁶<br/>(1-r)[1 - (2x + 4x² + 8x³ + 16x⁴ + 32x⁵)] = 1 - r⁶</p><p><strong>Step 5:</strong> Substitute the geometric series result:<br/>(1-r)[1 - 2x(1-32x⁵)/(1-2x)] = 1 - r⁶</p><p><strong>Step 6:</strong> Simplify the bracket:<br/>(1-r)[(1-2x - 2x + 64x⁶)/(1-2x)] = 1 - r⁶<br/>(1-r)[(1-2x - 2x(1-32x⁵))/(1-2x)] = 1 - r⁶</p><p><strong>Step 7:</strong> For this equation to hold with the form 1 - r⁶ = (1-r)(1+r+r²+r³+r⁴+r⁵), compare structures. Notice that if r = 2x, then 1 - r⁶ = (1-r)(1+r+r²+r³+r⁴+r⁵).</p><p><strong>Step 8:</strong> Comparing both sides, the geometric series 1 + 2x + 4x² + 8x³ + 16x⁴ corresponds to 1 + r + r² + r³ + r⁴ when r = 2x.</p><p><strong>Step 9:</strong> If r = 2x, then r/x = 2x/x = 2.</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B