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Which one of the following is the closed form for the generating function of the sequence {aₙ}_(n≥0) defined below? aₙ = { n+1, n is odd { 1, otherwise

1. x(1+x²)/(1-x²)² + 1/(1-x)
2. x(3-x²)/(1-x²)² + 1/(1-x)
3. 2x/(1-x²)² + 1/(1-x)
4. x/(1-x²)² + 1/(1-x)

Correct answer: x(1+x²)/(1-x²)² + 1/(1-x)

Solution

The correct option accurately represents the generating function by combining the contributions from the odd-indexed terms, which follow the pattern of n+1, and the even-indexed terms, which are constant at 1. The use of the generating function formula for sequences allows for the correct summation of these two distinct behaviors.

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