Question

If the coefficient of \( x^r \) in the expansion of \( (1 + x + x^2)^{100} \) is \( a_r \), and \( S = \sum\limits_{r=0}^{300} a_r \), then

\[ \sum\limits_{r=0}^{300} r a_r = \]

• A. \( (50) S \)
• B. \( (25) S \)
• C. \( (150) S \)
• D. \( (100) S \)

Hint:

For coefficient summation problems involving polynomial expansions, use function evaluation techniques and derivative properties to determine weighted sums.

Answer 1

The Correct Option is C

Step 1: Understanding the Given Expansion 
The given function is: \[ f(x) = (1 + x + x^2)^{100}. \] The coefficient of \( x^r \) in the expansion of this expression is denoted as \( a_r \), meaning: \[ S = \sum\limits_{r=0}^{300} a_r = f(1). \] 

Step 2: Finding \( S \) 
To determine \( S \), we evaluate the function at \( x = 1 \): \[ S = f(1) = (1 + 1 + 1)^{100} = 3^{100}. \]

 Step 3: Computing \( \sum r a_r \) 
By differentiation, \[ f'(x) = 100(1 + x + x^2)^{99} \cdot (1 + 2x). \] Setting \( x = 1 \), \[ \sum\limits_{r=0}^{300} r a_r = f'(1). \] Substituting \( x = 1 \) into \( f'(x) \): \[ f'(1) = 100 (3^{99}) \cdot (1 + 2) = 100 \times 3^{99} \times 3 = 300 \times 3^{99}. \] Thus, \[ \sum\limits_{r=0}^{300} r a_r = 150 S. \]