Math Question.......Thomas is training his pet frog to climb up a flight of stairs....Continued?

Question

Thomas is training his pet frog to climb up a flight of stairs. The frog can hop up only one or two steps, and can not hop down. How many different ways can the frog hop up a flight of 10 stairs? (up 1 up 2 up 1 up 2 up 1 up1 up1 up1 etc.)

WHAT IS THE METHOD OF SOLVING THIS PROBLEM??????

thanks

Answer 1

A good technique is to start with a smaller number of steps and look for a pattern.

For 1 step, there is 1 way: 1
For 2 steps, there are 2 ways: 1,1; 2
For 3 steps, there are 3 ways: 1,1,1; 1,2; 2,1
...
After doing this a few more times, try to make a general formula. Let s(n) = number of ways to go up n steps. Notice that if the first hop is 1 step then there are s(n-1) ways to go up the remaining n-1 steps. Similarly, if the first hop is 2 steps then there are s(n-2) ways to go up the remaining n-2 steps. So we get the recursive formula:

s(n) = s(n-1) + s(n-2)
thus s(n) is the nth fibonacci number (or the n+1st depending upon when you start counting), so s(10) = 89

Answer 2

it's the exponetial problem, there 2 way frog can leap, either 1 step or 2 step, and there are 10 steps total; therefore,

2^10 ways or 1024 ways

Answer 3

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Answer 4

Hi. Two possibilities for the first step, two for the second, third etc.