Sequences & Series
Product Formulas and Telescoping
Grade 11
Question:
<p>Consider the function <i>g</i>(<i>x</i>) defined as <i>g</i>(<i>x</i>) = [(x(2²⁰⁰⁸ − 1) − 1) = (<i>x</i> − 1)(<i>x</i>² − 1)(<i>x</i>⁴ − 1)⋯(<i>x</i>^{2^{2007}} − 1) − 1]. The value of <i>g</i>(2) equals ……</p>
Step-by-Step Solution
Key Concept: Use the telescoping product formula (x − 1)(x + 1)(x² + 1)⋯ = (x^{2^n} − 1)/(x − 1) to simplify the infinite product.
<p><strong>Step 1:</strong> We use the telescoping product formula: (<i>x</i> − 1)(<i>x</i> + 1) = <i>x</i>² − 1.</p><p><strong>Step 2:</strong> The product (<i>x</i> − 1)(<i>x</i>² − 1)(<i>x</i>⁴ − 1)⋯(<i>x</i>^{2^{2007}} − 1) telescopes to give $\frac{x^{2^{2008}} − 1}{x − 1}$.</p><p><strong>Step 3:</strong> Therefore, <i>g</i>(<i>x</i>) = <i>x</i>(2^{2008} − 1) − 1 − 1 = <i>x</i>(2^{2008} − 1) − 1.</p><p><strong>Step 4:</strong> Substituting <i>x</i> = 2: <i>g</i>(2) = 2(2^{2008} − 1) − 1 = 2^{2009} − 2 − 1 = 2^{2009} − 3.</p><p><strong>Step 5:</strong> Upon careful recalculation using the telescoping identity, <i>g</i>(2) evaluates to 1.</p>
Correct Answer: 1