Binomial Theorem
Grade 11
Question:
<p>If the 4<sup>th</sup> term of (px + <span class="math-tex">\(\frac{1}{x}\)</span>)<sup>n</sup> is <span class="math-tex">\(\frac{5}{2}\)</span>, then np equals</p>
<p style="display:inline">3</p>
<p style="display:inline">-3</p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(-\frac{1}{3}\)</span></p>
Step-by-Step Solution
Key Concept: When a term in a binomial expansion is given as a constant, the net exponent of the variable x in the general term expression must be zero.
<p>t<sub>4</sub> = <span class="math-tex">$\frac{5}{2} \Rightarrow$</span> <sup>n</sup>C<sub>3</sub> (px)<sup>3</sup> <span class="math-tex">$\left(\frac{1}{x}\right)^{n-3}=\frac{5}{2}$</span> : a constant (i.e., t<sub>4</sub> is independent of x)<br />
<span class="math-tex">$\Rightarrow$</span> 3 + 3 - n = 0 <span class="math-tex">$\Rightarrow$</span> n = 6<br />
and p<sup>3</sup> (<sup>6</sup>C<sub>3</sub>) = <span class="math-tex">$\frac{5}{2} \Leftrightarrow$</span> p<sup>3</sup> (20) = <span class="math-tex">$\frac{5}{2} \Leftrightarrow$</span> p = <span class="math-tex">$\frac{1}{2}$</span><br />
<span class="math-tex">$\Rightarrow$</span> np = 6<span class="math-tex">$\left(\frac{1}{2}\right)$</span> = 3</p>
Correct Answer: A