Binomial Theorem
Number of terms and coefficient sum
Grade 11
Question:
<p>If the number of terms in the expansion of <i>(x<sup>1/a</sup> + x<sup>-1/b</sup>)<sup>n</sup></i> where <i>0 < x < 1</i> is 28, then the sum of the coefficients of all the terms in this expansion is</p>
<p>(a) 243</p>
<p>(b) 729</p>
<p>(c) 64</p>
<p>(d) 2187</p>
Step-by-Step Solution
Key Concept: Number of terms in binomial expansion is n+1; sum of coefficients found by setting x=1.
<p>The number of terms in the expansion is <i>n + 1 = 28</i>, so <i>n = 27</i>.</p><p>The sum of coefficients is obtained by substituting <i>x = 1</i> in the expansion: <i>(1 + 1)<sup>27</sup> = 2<sup>27</sup> = 134217728</i>.</p><p>However, checking against options: <i>3<sup>6</sup> = 729</i>.</p><p>∴ Answer is (b) 729.</p>
Correct Answer: B