Binomial Theorem
Grade None
Question:
<p>If C<sub>0</sub>, C<sub>1</sub>, C<sub>2</sub>, ..., C<sub>n</sub> are the binomial coefficients, then 2<span class="math-tex">\(\cdot\)</span>C<sub>1</sub> + 2<sup>3</sup><span class="math-tex">\(\cdot\)</span>C<sub>3</sub> + 2<sup>5</sup><span class="math-tex">\(\cdot\)</span>C<sub>5</sub> + ... equals</p>
<p style="display:inline"><span class="math-tex">\(\frac{3^{n}+(-1)^{n}}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{3^{n}+1}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{3^{n}-1}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{3^{n}-(-1)^{n}}{2}\)</span></p>
Step-by-Step Solution
Key Concept: Use the difference between the binomial expansions of (1+x)^n and (1-x)^n to isolate terms with odd indices and substitute the specific value of x.
<p>We know that,<br />
[(1 + x)<sup>n</sup> - (1 - x)<sup>n</sup>] = 2[C<sub>1</sub>x + C<sub>3</sub>x<sup>3</sup> + C<sub>5</sub>x<sup>5</sup> + ...]<br />
<span class="math-tex">$\Rightarrow \frac{1}{2}$</span> [(1 + x)<sup>n</sup> - (1 - x)<sup>n</sup>] = C<sub>1</sub>x + C<sub>3</sub>x<sup>3</sup> + C<sub>5</sub>x<sup>5</sup> + ...<br />
Substituting x = 2, we get,<br />
2<span class="math-tex">$\cdot$</span>C<sub>1</sub> + 2<sup>3</sup><span class="math-tex">$\cdot$</span>C<sub>3</sub> + 2<sup>5</sup><span class="math-tex">$\cdot$</span>C<sub>5</sub> + ... = <span class="math-tex">$\frac{3^{n}-(-1)^{n}}{2}$</span></p>
Correct Answer: D