Sequences
Overview
Sequences are ordered lists of numbers (called "terms"), like 2,5,8. Some sequences follow a specific pattern that can be used to extend them indefinitely. For example, 2,5,8 follows the pattern "add 3," and now we can continue the sequence. Sequences can have formulas that tell us how to find any term in the sequence.
Sequences
Sequences
Definition: A sequence is a set of numbers in a specific order. 2, 5, 8,…. is an example of a sequence.
Note that a sequence may have either a finite or an infinite number of terms.
The terms of a sequence are the individual numbers in the sequence. If we let a1 represent the first term of a sequence, an represent the nth term, and n represent the term number, then the sequence is represented by a1, a2, a3, ….,an, … In the example above, a1=2, a2=5, a3= 8, etc.
Arithmetic Sequences
Definition: An arithmetic sequence is a sequence in which each term, after the first, is the sum of the preceding term and a common difference.
An arithmetic sequence can be represented by a1, a1 +d, a1 + 2d, …. In the sequence 2, 5, 8, ….. the common difference is 3.
The sequences 1, 3, 5, 7, ….. and 2, 8, 14, 20, ….. are examples of arithmetic sequences. Each has the property that the difference between any two immediate successive terms is constant.
The existence of a common difference is the characteristic feature of an arithmetic sequence. To test whether a given sequence is an arithmetic sequence, determine whether a common difference exists between every pair of successive terms. For example, 4, 8, 9, 16, 32, …. is not an arithmetic sequence because the difference between the first two terms is 4, but the difference between the second and third terms is 8.
If a1 is the first term of an arithmetic sequence, an the nth term, d is the common difference, a formula for finding the value of the nth term of an arithmetic sequence is:
an = a1 + (n – 1)d
The formula for the nth term of an arithmetic sequence may be used to find any term of the sequence. This is done by choosing the appropriate value of n and substituting in the formula above.
For example, find the 75th term of the sequence 2, 5, 8,……
a75 = a1 + (n – 1)d.
Since a1 = 2, n = 75, d = 3, then
a75 = 2 + (75 – 1)(3)
= 2 + (74)(3)
= 2 + 222
= 224
Thus, a75 = 222.