Definition: An explicit formula gives the nth term of a sequence directly from its position n, so any term can be computed without finding the terms before it.
What is an explicit formula?
It is a function of the term number. Put in n, get out an. You can reach the 500th term in one calculation.
What are the standard forms?
Arithmetic: an = a1 + (n – 1)d. Geometric: an = a1 · rn-1. Other patterns, such as squares, have their own rules, like an = n2.
Example. The sequence 5, 8, 11, 14 has a1 = 5 and d = 3, so an = 5 + (n – 1)(3) = 3n + 2. Check: a4 = 3(4) + 2 = 14. The 80th term is 3(80) + 2 = 242.
What should you check?
Always test your formula on the first one or two terms. A very common slip is to write an = 5 + 3n, which gives 8 for the first term instead of 5. The explicit and recursive forms must describe the same list; if they disagree, one has an error. The domain is the positive integers only, so fractional values of n do not make sense.
Where to learn it
- Arithmetic and Geometric Sequences as Functions – lesson
- Building Functions from Real Situations – lesson