Question

If \( C_0, C_1, C_2, \dots \) denote the binomial coefficients in the expansion of \( (1 + x)^n \), then the value of \[ C_0 + (C_0 + C_1) + (C_0 + C_1 + C_2) + \dots + (C_0 + C_1 + C_2 + \dots + C_n) \] is

• A. \( n^2 \)
• B. \( (n-1)2^n \)
• C. \( (n+1)2^n \)
• D. \( n2^n \)

Hint:

The sum of the binomial coefficients for a given power of \( n \) is \( 2^n \), and the sum of partial sums increases by a factor of \( (n+1) \).

Answer 1

The Correct Option is C

Step 1: Use the sum of binomial coefficients.
The sum of binomial coefficients up to \( C_n \) for \( (1 + x)^n \) is \( 2^n \). Therefore, the sum of the terms in the given series is \( (n+1) \times 2^n \).
Step 2: Conclusion.
The required sum is \( (n+1)2^n \). Final Answer: \[ \boxed{(n+1)2^n} \]