Binomial Theorem
Grade None
Question:
<p>If (1 + x)<sup>n</sup> = C<sub>0</sub> + C<sub>1</sub>x + C<sub>2</sub>x<sup>2</sup> + ... + C<sub>n</sub>x<sup>n</sup>, then <span class="math-tex">\(\frac{C_{1}}{C_{0}}+\frac{2 C_{2}}{C_{1}}+\frac{3 C_{3}}{C_{2}}+\ldots \frac{n C_{n}}{C_{n-1}}\)</span> =</p>
<p style="display:inline"><span class="math-tex">\(\frac{n(n+2)}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{n(n+1)}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{(n-1)(n-2)}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{n(n-1)}{2}\)</span></p>
Step-by-Step Solution
Key Concept: Simplify the series by applying the binomial coefficient ratio property $\frac{C_r}{C_{r-1}} = \frac{n-r+1}{r}$, which transforms the general term $r \frac{C_r}{C_{r-1}}$ into $n-r+1$.
<p><span class="math-tex">$\frac{C_{1}}{C_{0}}+2 \cdot \frac{C_{2}}{C_{1}}+3 \cdot \frac{C_{2}}{C_{2}}+\ldots n \cdot \frac{C_{n}}{C_{n-1}}$</span><br />
<span class="math-tex">$=\frac{n}{1}+2 \frac{\frac{n(n-1)}{1.2}}{n}+3 \frac{\frac{n(n-1)(n-2)}{3.2 .1}}{\frac{n(n-1)}{1.2}}+\ldots n \frac{1}{n}$</span><br />
= n + (n - 1) + (n - 2) + ... + 1<br />
= <span class="math-tex">$\sum$</span>n<br />
= <span class="math-tex">$\frac{n(n+1)}{2}$</span></p>
Correct Answer: B