The first sequence of differences would be:ģa+b, 5a+b, 7a+b, \dots 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. This sequence was created by plugging in 1 for n, 2 for n, 3 for n, etc. Then the sequence would look like:Ī+b+c, 4a+2b+c, 9a+3b+c, \dots Suppose the nth term of a sequence was given by an^2+bn+c. So, the nth term of the sequence is given by 2n+3. Using the first number in the sequence and the first term: It is possible to solve for b using one of the terms in the sequence. The difference between each term and the term after it is a. ![]() Then the sequence looks like:Ī+b, 2a+b, 3a+b, \dots Suppose the formula for the sequence is given by an+b for some constants a and b. Since this difference is constant, and this is the first set of differences, the sequence is given by a first-degree (linear) polynomial. ![]() In fact, the difference between each pair of terms is 2. The difference between 7 and 9 is also 2. If a sequence is generated by a polynomial, this fact can be detected by noticing whether the computed differences eventually become constant.ĥ, 7, 9, 11, 13, \dots Such a formula will produce the nth term when a value for the integer n is put into the formula. Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence. general term: A mathematical expression containing variables and constants that, when substituting integer values for each variable, produces a valid term in a sequence.sequence: A set of things next to each other in a set order a series.Once a constant difference is achieved, one can solve equations to generate the formula for the polynomial.If the differences eventually become constant, then the sequence is generated by a polynomial formula. By hand, one can take the differences between each term, then the differences between the differences in terms, etc.Given terms in a sequence generated by a polynomial, there is a method to determine the formula for the polynomial.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. ![]() So the nth term can be described by the formula a_n = a_ An arithmetic sequence is one in which a term is obtained by adding a constant to a previous term of a sequence.Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence. The number of ordered elements (possibly infinite ) is called the length of the sequence.
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