| A Recursive Partitioned Function |
Consider a function, f(n), defined recursively by parts:
f(n+1) = f(n) + 3 while f(n) <= 100
f(n+1) = f(n) - 2 while f(n) > 100
f(1) = 50
Find the value of f(500). |
Problem Moderated by: Sasha |
| Problem Solution |
Our function won't be very interesting until it approaches 100, so let's find out when it does this.
f(1) = 50
For a while, f(n) = 50 + 3(n-1)
So,
f(17) = 98
f(18) = 101
f(19) = 99
f(20) = 102
f(21) = 100
f(22) = 103
f(23) = 101
Now, a pattern has been established: f(a+5k) = f(a) while a>=18 and k is a whole number.
f(500) = f(20+5(96)) = f(20) = 102 |
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