An explicit formula describes the nth term of a sequence using the number n.
For example, in the sequence 2, 4, 6, 8, 10, …, the nth term is twice the value of n. You write this as
| n | nth term |
|---|---|
| 1 |
|
| 2 |
|
| 3 |
|
| 4 |
|
Problem 1 Generating a Sequence Using an Explicit Formula
A sequence has an explicit formula
Plan
How does the explicit formula help you find the value of a term?
Replace n in the formula with the number of the term. Simplify to find the value of the term.
You can use a table to organize your work for the remaining terms.
| n |
|
|
|---|---|---|
| 3 |
|
Substitute 3 for n and simplify. |
| 4 |
|
|
| 5 |
|
And so on. |
| 6 |
|
|
| 7 |
|
|
| 8 |
|
|
| 9 |
|
|
| 10 |
|
The first ten terms are 1, 4, 7, 10, 13, 16, 19, 22, 25, 28.
✓ Got It?
- A sequence has an explicit formula
Sometimes you can see the pattern in a sequence by comparing each term to the one that came before it. For example, in the sequence 133, 130, 127, 124, …, each term after the first term is equal to three less than the previous term.
A recursive definition for this sequence contains two parts.
- an initial condition (the value of the first term):
- a recursive formula (relates each term after the first term to the one before it):
Table of Contents
- 5-1 Polynomial Functions
- 5-2 Polynomials, Linear Factors, and Zeros
- 5-3 Solving Polynomial Equations
- 5-4 Dividing Polynomials
- 5-5 Theorems About Roots of Polynomial Equations
- 5-6 The Fundamental Theorem of Algebra
- 5-7 The Binomial Theorem
- 5-8 Polynomial Models in the Real World
- 5-9 Transforming Polynomial Functions