The next term in the sequence is obtained by subtracting 4 from the previous term:
13 = 17 - 4
9 = 13 - 4
5 = 9 - 4
and so on.
Recursively, we can describe the n-th term in the sequence (denoted a(n)) as a function of the (n - 1)-th term:
a(n) = a(n - 1) - 4
starting with a(1) = 17. So we have, for instance,
a(2) = a(1) - 4 = 17 - 4 = 13
We want to find an explicit formula for a(n), meaning we want a way to determine the n-th term only as a function of n, and not any other known term in the sequence. We can iteratively apply the recursive rule above:
a(n) = a(n - 1) - 4
a(n - 1) = a(n - 2) - 4
→ a(n) = (a(n - 2) - 4) - 4 = a(n - 2) - 2×4
a(n - 2) = a(n - 3) - 4
→ a(n) = (a(n - 3) - 4) - 2×4 = a(n - 3) - 3×4
a(n - 3) = a(n - 4) - 4
→ a(n) = (a(n - 4) - 4) - 3×4 = a(n - 4) - 4×4
and so on, down to
a(n) = a(1) - (n - 1)×4
The first term is a(1) = 17, so
a(n) = 17 - 4 (n - 1) = 21 - 4n
