Sequences and Summation Notation

Definition of a Sequence An infinite sequence {an} is a function whose domain is the set of positive integers. The function values, or terms, of the sequence are represented by a1, a2, a3, a4,…, an ,…. Sequences whose domains consist only of the first n positive integers are called finite sequences.

Text Example Write the first four terms of the sequence whose nth term, or general term, is given: an = 3n + 4. Solution We need to find the first four terms of the sequence whose general term is an = 3n + 4. To do so, we replace each occurrence of n in the formula by 1, 2, 3, and 4. 3 · 1 + 4 = 3 + 4 = 7 a1, 1st term 3 · 2 + 4 = 6 + 4 = 10 a2, 2nd term 3 · 3 + 4 = 9 + 4 = 13 a3, 3rd term 3 · 4 + 4 = 12 + 4 = 16 a4, 4th term The first four terms are 7, 10, 13, and 16. The sequence defined by an = 3n + 4 can be written as 7, 10, 13, …, 3n + 4, ….

Factorial Notation If n is a positive integer, the notation n! is the product of all positive integers from n down through 1. n! = n(n-1)(n-2)…(3)(2)(1) 0! , by definition is 1.

Summation Notation The sum of the first n terms of a sequence is represented by the summation notation Where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

Example Expand and evaluate the sum: Solution:

Example Express the sum using summation notation: Solution:

Example Express the sum using summation notation: Solution:

Sequences and Summation Notation