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Evaluate the sum of binomial coefficients with indices divisible by 4: C(4n, 0) + C(4n, 4) + C(4n, 8) + ⋯ + C(4n, 4n).

1. (-1)ⁿ 2²ⁿ⁻¹ + 2⁴ⁿ⁻²
2. (-1)ⁿ 2²ⁿ⁻¹
3. (-1)ⁿ 2⁴ⁿ⁻¹
4. None of these

Correct answer: (-1)ⁿ 2²ⁿ⁻¹ + 2⁴ⁿ⁻²

Solution

Sum over k divisible by 4 = (1/4)[(1+1)^4n+(1+i)^4n+(1-1)^4n+(1-i)^4n]. Since (1+i)^4n=(1-i)^4n=2^(2n)(-1)^n, the sum = 2^(4n-2)+(-1)^n*2^(2n-1).

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