Answer:
(a) [tex]x_{n} = x_{n - 2} + x_{n - 4} + x_{n - 8}[/tex]
(b) 18 ways
Solution:
As per the question:
We need to find the no.of ways to go 'n' miles by foot:
From the question it is clear that, at each hours even miles can be traveled by a person, If the distance to be covered is odd, then there in no way 3 miles can be covered.
Thus defining recurrence as:
[tex]x_{o} = 1[/tex]
[tex]x_{n} = 0[/tex]
where
n: negative or odd
Then
[tex]x_{n} = x_{n - 2} + x_{n - 4} + x_{n - 8}[/tex]
(b) No. of ways to travel 12 miles:
[tex]x_{o} = 1[/tex]
[tex]x_{2} = 1 + 0 = 1[/tex]
[tex]x_{4} = x_{2} + x_{o} + 0 = 1 + 1 = 2[/tex]
[tex]x_{6} = x_{4} + x_{2} + 0 = 2 + 1 + 0 = 3[/tex]
[tex]x_{8} = x_{6} + x_{4} + 0 = 3 + 2 = 6[/tex]
[tex]x_{10} = x_{8} + x_{6} + x_{2} = 6 + 3 + 1 = 10[/tex]
[tex]x_{12} = x_{10} + x_{8} + x_{4} = 10 + 6 + 2 = 18[/tex]
Thus clearly there are 18 ways to travel 12 miles
