Question

Mr. Juan is used to going up the stairs in his building, jumping 1 to 2 steps, as he sees fit. Determine the number of ways that Mr. John can climb the 20-step ladder, using the "Z" transform.

Answer 1

The number of ways Mr. Juan can climb a 20-step ladder using jumps of 1 or 2 steps, found via the "Z" transform, is the coefficient of
`\( z^(20) \)` in the function
`\( F(z) = (1)/(1 - z - z^2) \)`. Calculating this coefficient gives the total count of ways to ascend the ladder.

Step-by-step explanation:

Using the "Z" transform for counting the number of ways to climb the ladder involves creating a generating function that represents the possible combinations of jumps. Considering that Mr. Juan can jump either 1 or 2 steps at a time, the number of ways to climb the ladder follows a specific pattern. Let the function that represents this scenario be denoted as F(z).

The equation representing the situation is F(z) = 1 / (1 - z - z²), where the denominator reflects the options available for Mr. Juan to move up the ladder—either taking a single step (z) or a double step (z²). The 1 in the numerator signifies the starting point.

To determine the number of ways Mr. Juan can climb the 20-step ladder, calculate the coefficient of z²⁰ in the function F(z). This involves expanding F(z) and isolating the term with z²⁰. By solving for this coefficient, the specific count of ways Mr. Juan can reach the 20th step using the given jump options can be found. Calculations involve polynomial expansion and simplification to extract the coefficient corresponding to z²⁰.

Using this approach allows us to efficiently count the various combinations Mr. Juan can take to ascend the 20-step ladder, encapsulating all possible sequences of 1 and 2 step jumps that lead to reaching the 20th step.