Question

If a staircase has six steps and Lucas can advance by either one or two steps at a time, how many different ways can Lucas walk up the staircase starting from the ground?

Answer 1

The number of different ways Lucas can walk up a six-step staircase, advancing by one or two steps at a time, is 13. This is calculated using a method similar to the Fibonacci sequence.

Step-by-step explanation:

The question at hand involves determining the number of different ways Lucas can walk up a staircase with six steps, given he can advance by either one or two steps at a time. To solve this problem, we can use a method similar to the Fibonacci sequence, where each step number is the sum of the ways to reach the previous two steps. Let's denote the ways to reach step i as F(i).

• For the first step (i.e., step 1), Lucas has only one way to reach it, which is one step (F(1) = 1).
• For the second step (i.e., step 2), Lucas can either take two one-steps or one two-step (F(2) = 2).
• For the third step (i.e., step 3), Lucas can reach it from the first step with two steps or from the second step with one step (F(3) = F(2) + F(1) = 2 + 1 = 3).
• Following this pattern, we calculate the number of ways to reach each subsequent step:

1. F(4) = F(3) + F(2) = 3 + 2 = 5
2. F(5) = F(4) + F(3) = 5 + 3 = 8
3. F(6) = F(5) + F(4) = 8 + 5 = 13

Therefore, there are 13 different ways Lucas can walk up the six-step staircase.