The General Term of a Sequence
Given terms in a sequence, it is often possible to find a formula for the general term of the sequence, if the formula is a polynomial.
Learning Objective

Practice finding a formula for the general term of a sequence
Key Points
 Given terms in a sequence generated by a polynomial, there is a method to determine the formula for the polynomial.
 By hand, one can take the differences between each term, then the differences between the differences in terms, etc. If the differences eventually become constant, then the sequence is generated by a polynomial formula.
 Once a constant difference is achieved, one can solve equations to generate the formula for the polynomial.
Terms

sequence
A set of things next to each other in a set order; a series

general term
A mathematical expression containing variables and constants that, when substituting integer values for each variable, produces a valid term in a sequence.
Full Text
Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence. Such a formula will produce the
If a sequence is generated by a polynomial, this fact can be detected by noticing whether the computed differences eventually become constant.
Linear Polynomials
Consider the sequence:
The difference between
Suppose the formula for the sequence is given by
The difference between each term and the term after it is
So, the
Quadratic Polynomials
Suppose the
This sequence was created by plugging in
If we start at the second term, and subtract the previous term from each term in the sequence, we can get a new sequence made up of the differences between terms. The first sequence of differences would be:
Now, we take the differences between terms in the new sequence. The second sequence of differences is:
The computed differences have converged to a constant after the second sequence of differences. This means that it was a secondorder (quadratic) sequence. Working backward from this, we could find the general term for any quadratic sequence.
Example
Consider the sequence:
The difference between
This list is still not constant. However, finding the difference between terms once more, we get:
This fact tells us that there is a polynomial formula describing our sequence. Since we had to do differences twice, it is a seconddegree (quadratic) polynomial.
We can find the formula by realizing that the constant term is
Next we note that the first item in our first list of differences is
Finally, note that the first term in the sequence is
So,
General Polynomial Sequences
This method of finding differences can be extended to find the general term of a polynomial sequence of any order. For higher orders, it will take more rounds of taking differences for the differences to become constant, and more backsubstitution will be necessary in order to solve for the general term.
General Terms of NonPolynomial Sequences
Some sequences are generated by a general term which is not a polynomial. For example, the geometric sequence
General terms of nonpolynomial sequences can be found by observation, as above, or by other means which are beyond our scope for now. Given any general term, the sequence can be generated by plugging in successive values of
Key Term Reference
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 difference
 Appears in these related concepts: Factoring a Difference of Squares, The Order of Operations, and Basic Operations
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 geometric
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Sums and Series, and Introduction to Sequences
 geometric sequence
 Appears in these related concepts: Summing the First n Terms in a Geometric Sequence, Recursive Definitions, and Geometric Sequences
 integer
 Appears in these related concepts: Scientific Notation, Binomial Expansions and Pascal's Triangle, and Finding a Specific Term
 linear
 Appears in these related concepts: Exponential Growth and Decay, Graphs of Linear Inequalities, and Factoring General Quadratics
 polynomial
 Appears in these related concepts: Domains of Rational and Radical Functions, Simplifying, Multiplying, and Dividing, and Partial Fractions
 quadratic
 Appears in these related concepts: The Discriminant, Stretching and Shrinking, and What is a Quadratic Function?
 set
 Appears in these related concepts: Sequences, Sets of Numbers, and Sequences of Mathematical Statements
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Introduction to Variables
Sources
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