Laura is training her pet white rabbit, Sugar, to climb up a flight of 25 steps. Sugar can hop up one or two steps at a time. She never hops back down, only up. How many different ways can Sugar hop up the flight of steps?
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To determine the number of different ways Sugar can hop up the flight of 25 steps, we can use a dynamic programming approach. Let's break it down step by step:
Step 1: Identify the base cases
- If the number of steps is 0, there is only one way to reach the top (by not moving at all).
- If the number of steps is 1, there is only one way to reach the top (by hopping up one step).
Step 2: Define the recursive relationship
- Sugar can hop up either one step or two steps at a time. So, the number of ways to reach the top of a flight of n steps can be calculated by adding the number of ways to reach step n-1 and the number of ways to reach step n-2.
Step 3: Implement the dynamic programming approach
- We can use an array to store and update the number of ways to reach each step.
- Initialize two variables, `ways[n]` and `ways[n+1]` to 0.
- Iterate from the bottom to the top, updating the number of ways to reach each step based on the recursive relationship.
Here's the step-by-step breakdown of the implementation:
```
// Initialize the number of ways to reach step 0 and 1
ways = [1, 1]
// Iterate from step 2 to step 25
for i in range(2, 26):
ways.append(ways[i-1] + ways[i-2])
// The number of ways to reach the top of a flight of 25 steps will be at index 25 of the `ways` array
result = ways[25]
```
So, the number of different ways Sugar can hop up the flight of 25 steps is 75025.
I hope this helps! Let me know if you have any further questions.
To find the number of different ways Sugar can hop up the flight of 25 steps, we'll use a dynamic programming approach.
Let's first consider the base cases:
- If there are 0 steps, there is only one way to climb up (by doing nothing).
- If there is 1 step, there is only one way to climb up (by hopping up one step).
When there are more steps, we can consider the number of ways to hop up the previous two steps. Since Sugar can only hop up one or two steps at a time, the total number of ways to climb up the current step is equal to the sum of the number of ways to climb up the previous step and the number of ways to climb up the step before the previous one.
Based on this logic, we can create an array to store the number of ways to climb up each step. We'll start by initializing the first two elements as 1, since we know the base cases. Then, for each step from 2 to 25, we'll calculate the number of ways to climb up that step using the formula explained above.
Here is the Python code to calculate the number of ways:
```python
# Initialize array to store number of ways to climb up each step
ways = [0] * (25 + 1)
ways[0] = 1
ways[1] = 1
# Calculate number of ways for each step
for i in range(2, 25 + 1):
ways[i] = ways[i - 1] + ways[i - 2]
# Print the number of ways to climb up the flight of 25 steps
print(ways[25])
```
This code will output the number of different ways Sugar can hop up the flight of steps, which is the answer to your question.