Binomial Theorem
Number of terms and sum of coefficients
Grade 11
Question:
<p>If the number of terms in the expansion of \(\left(1 - \dfrac{2}{x} + \dfrac{4}{x^2}\right)^n\), \(x \neq 0\), is 28, then the sum of the coefficients of all the terms in this expansion, is</p>
<p>2187</p>
<p>243</p>
<p>729</p>
<p>64</p>
Step-by-Step Solution
Key Concept: The number of terms in a multinomial expansion of (a + b + c)^n is C(n+2, 2) = (n+1)(n+2)/2. Once n is found, substitute x = 1 to find the sum of all coefficients.
<p><strong>Step 1: Find the value of n using the number of terms formula.</strong></p><p>For the expansion of (1 - 2/x + 4/x²)^n, we have three distinct terms: 1, -2/x, and 4/x².</p><p>The number of terms in the expansion of (a + b + c)^n is given by C(n+2, 2) = (n+1)(n+2)/2</p><p>Given: Number of terms = 28</p><p>Therefore: (n+1)(n+2)/2 = 28</p><p>(n+1)(n+2) = 56</p><p><strong>Step 2: Solve the quadratic equation.</strong></p><p>n² + 3n + 2 = 56</p><p>n² + 3n - 54 = 0</p><p>Using factorization: (n + 9)(n - 6) = 0</p><p>Since n must be positive: n = 6</p><p><strong>Step 3: Find the sum of all coefficients.</strong></p><p>To find the sum of all coefficients in the expansion, substitute x = 1 in the original expression:</p><p>Sum = (1 - 2/1 + 4/1²)^6</p><p>Sum = (1 - 2 + 4)^6</p><p>Sum = (3)^6</p><p>Sum = 729</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C