Suppose that for any integer n,

f(n)= (n-1). if n is even;
f(n)= (2n). if n is odd.
If k ∈ N, and f (f (f(k))) = 21, find the sum of the digits in k

I really don't get this means. (∈)

"is an element of "

see:
http://www.glenoaks.edu/facultystaff/FacultyWebSites/JillToner/Documents/NSM105%20-%20Math%20for%20El%20Ed/Section%202.1%20-%20Set%20Theory.pdf

f [ f ( f(k) ) ] = 21

f ( f(k) ) = 22

f(k) = 11

k = 12

so about two digits in twelve

The symbol "∈" represents the mathematical concept of "an element of" or "belongs to". In the context of this problem, it means that k is a natural number or positive integer.

Now, let's try to solve the problem step by step:

Given f(n), we can see that it has two cases:
1. If n is even, f(n) = n-1.
2. If n is odd, f(n) = 2n.

Let's analyze f(f(f(k))) = 21:

1. Start with the innermost function f(k):
- If k is even, f(k) = k-1.
- If k is odd, f(k) = 2k.

2. Apply the middle function to f(k):
- If f(k) is even, f(f(k)) = f(k)-1 = (k-1)-1 = k-2.
- If f(k) is odd, f(f(k)) = f(k)*2 = (2k)*2 = 4k.

3. Apply the outermost function to f(f(k)):
- If f(f(k)) is even, f(f(f(k))) = f(f(k))-1 = (k-2)-1 = k-3.
- If f(f(k)) is odd, f(f(f(k))) = f(f(k))*2 = (4k)*2 = 8k.

We are given that f(f(f(k))) = 21, so we equate that to the two cases:

1. If f(f(f(k))) = k-3 = 21, solve for k:
k = 21 + 3 = 24.

2. If f(f(f(k))) = 8k = 21, solve for k:
k = 21/8 = 2.625.

However, k has to be a natural number or positive integer, so the solution is k = 24.

Now, to find the sum of the digits in k, we add up all the digits:
k = 24
Sum of digits = 2 + 4 = 6.

Therefore, the sum of the digits in k is 6.