Binomial Theorem
Series Summation
Grade 11
Question:
<p>The series <math>{}^nC_1 + 1 \times {}^nC_2 + 2 \times {}^nC_3 + \cdots + n \times {}^nC_n</math> is equal to</p>
<p>(a) <math>n2^{n-1}</math></p>
<p>(b) <math>2^{n-1}</math></p>
<p>(c) <math>n2^n</math></p>
<p>(d) <math>2^{3-1}</math></p>
Step-by-Step Solution
Key Concept: Differentiate the binomial expansion of (1+x)^n and substitute x=1 to find the sum of the series involving binomial coefficients with linear coefficients.
<p><strong>Solution:</strong></p><p>We know that <math>(1+x)^n = {}^nC_0 + {}^nC_1 x + {}^nC_2 x^2 + \cdots + {}^nC_n x^n</math></p><p>On differentiating both sides with respect to <math>x</math>:</p><p><math>n(1+x)^{n-1} = {}^nC_1 + 2 \times {}^nC_2 x + 3 \times {}^nC_3 x^2 + \cdots + n \times {}^nC_n x^{n-1}</math></p><p>Putting <math>x = 1</math>:</p><p><math>n(1+1)^{n-1} = {}^nC_1 + 2 \times {}^nC_2 + 3 \times {}^nC_3 + \cdots + n \times {}^nC_n</math></p><p><math>n \cdot 2^{n-1} = {}^nC_1 + 1 \times {}^nC_2 + 2 \times {}^nC_3 + \cdots + n \times {}^nC_n</math></p><p>∴ Answer is <math>n2^{n-1}</math></p>
Correct Answer: a