Binomial Theorem
Grade None
Question:
<p>If the sum of coefficients in the expansion of (1 + x<sup>3</sup>)<sup>n</sup> is p, that of (1 - 4x + 7x<sup>2</sup>)<sup>n</sup> is q and (1 - 5x + 10x<sup>2</sup> + 2x<sup>3</sup>)<sup>n</sup> is r, then</p>
<p style="display:inline">p, q, r are in G.P.</p>
<p style="display:inline">p, q, r are in A.P.</p>
<p style="display:inline">p = qr</p>
<p style="display:inline">p<sup>2</sup> = qr<sup>3</sup></p>
Step-by-Step Solution
Key Concept: The sum of coefficients in any polynomial expansion is found by substituting all variables with the value 1.
<p>Sum of coefficients can be obtained by substituting x = 1<br />
<span class="math-tex">$\Rightarrow$</span> p = (1 + 1<sup>3</sup>)<sup>n</sup> = 2<sup>n</sup>, q = (1 - 4 + 7)<sup>n</sup> = 4<sup>n</sup> = 2<sup>2n</sup> and r = (1 - 5 + 10 + 2)<sup>n</sup> = 8<sup>n</sup> = 2<sup>3n</sup><br />
<span class="math-tex">$\Rightarrow$</span> p<span class="math-tex">$\cdot$</span>r = 2<sup>n</sup> <span class="math-tex">$\cdot$</span> 2<sup>3n</sup> = 2<sup>4n</sup> = (2<sup>2n</sup>)<sup>2</sup> = q<sup>2</sup><br />
<span class="math-tex">$\Rightarrow$</span> p, q, r are in G.P.</p>
Correct Answer: A