Respuesta :
Answer:
a) [tex] a_{n+2} = a_{n+1} + a_n [/tex]
b) a₁ = 1, a₂ = 2
3) a₈ = 34
Where aₙ counts the total ways to climb n stairs.
Step-by-step explanation:
We can obtain the total number of possibilities to climb n stairs by using a recurrence relation. Since you can take 1 or 2 stairs at the time, it would be logical to use two initial conditions. Lets call an the total number of possibilities to climb n stairs. We have:
To climb 1 stair, we have only 1 possibility.
To climb 2 stairs we have 2 possibilities, taking the 2 at the same time, or taking one at the time.
If n is eqaual or greater than 1, then n+2 will be equal or greater than 3, and we should be able to deduce an+2 from an and an+1. First, we have 2 possibilities: we climb 1 stair and then climb n+1, or we climb 2 stairs at the same time and then climb n. Since we have to pick one option, we obtain that the total amount of possibilities to climb n+2 stairs is the sum of an+1 with an. Therefore
a) [tex] a_{n+2} = a_{n+1} + a_n [/tex]
b) We obtained this before, [tex]a_1 = 1[/tex], [tex]a_2 = 2 . [/tex]
c) Lets calculate each term
a₁ = 1
a₂ = 2
a₃ = a₁+a₂ = 3
a₄ = a₃+a₂ = 3+2 = 5
a₅ = 5+3 = 8
a₆ = 8+5 = 13
a₇ = 13+8 = 21
a₈ = 21+13 = 34
Thus, there are 34 ways to climb 8 stairs.
