Question

1. In your own words, explain the difference between a recursive formula and a explicit formula. (Hint: USE THE NOTES to help you. *Page 10*)

2. Is the following sequence a recursive or explicit formula? Why?

an = a (n-1) - 2/3
a1 = 12/3

Answer 1

1. A recursive formula defines each term using one or more previous term(s) (and needs one or more starting values). An explicit formula gives the nth term directly as a function of n, so you can compute any term without knowing earlier terms.

Example:
- Recursive: a_n = a_{n-1} + 3, a_1 = 2
- Explicit: a_n = 2 + 3(n−1)

2. an = a(n−1) − 2/3 with a1 = 12/3 is a recursive formula because it defines a_n in terms of the previous term a_{n−1} and supplies a starting value. (Note: a1 = 12/3 = 4.)

Since the difference is constant (−2/3), it’s an arithmetic sequence. The explicit form is
a_n = 4 − (2/3)(n−1),
which can also be written a_n = (14 − 2n)/3 or a_n = (2(7−n))/3.