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If the expansion of (1 + x)ⁿ is written as ∑_(r=0)ⁿ C_r x^r, then the value of C₀ + (C₀ + C₁) + (C₀ + C₁ + C₂) + ⋯ + (C₀ + C₁ + C₂ + ⋯ + Cₙ) is

1. n·2^(n-1)
2. (n + 2)·2ⁿ
3. 2ⁿ
4. (n + 2)·2^(n-1)

Correct answer: (n + 2)·2^(n-1)

Solution

The sum = sum_(r=0)^n (n-r+1)C_r = (n+1)*2^n - n*2^(n-1) = 2^(n-1)[2(n+1)-n] = (n+2)*2^(n-1).

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