Binomial Theorem
Binomial Expansion
Grade 11
Question:
<p>Consider the binomial expansion <i>R = (1 + 2<i>x</i>)<sup><i>n</i></sup> = <i>I</i> + <i>f</i></i>, where <i>I</i> is the integral part of <i>R</i> and <i>f</i> is the fractional part of <i>R</i>, <i>n ∈ ℕ</i>. Also, the sum of coefficients of <i>R</i> is 2187. The value of <i>(n + Rf)</i> for <i>x = 1/2</i> is</p>
<p>(a) 7</p>
<p>(b) 8</p>
<p>(c) 9</p>
<p>(d) 10</p>
Step-by-Step Solution
Key Concept: Find <i>n</i> from sum of coefficients, then evaluate <i>R</i> at the given value of <i>x</i>.
<p><strong>Solution:</strong></p><p>Sum of coefficients: Set <i>x = 1</i>, then <i>(1+2)<sup><i>n</i></sup> = 3<sup><i>n</i></sup> = 2187 = 3<sup>7</sup></i>, so <i>n = 7</i>.</p><p>For <i>x = 1/2</i>: <i>R = (1 + 2·(1/2))<sup>7</sup> = 2<sup>7</sup> = 128 = I</i>, so <i>f = 0</i>.</p><p><i>n + Rf = 7 + 128·0 = 7</i>... (requires verification)</p>
Correct Answer: c