Series and Sequences An infinite sequence is an unending list of numbers that follow a pattern. The terms of the sequence are written a1, a2, a3,...,an,... If the list ends, we call it a finite sequence. 1 1 1

Ex. Write the first four terms of the sequence: a) an = 3n – 2 b) an = 3 + (-1)n

Ex. Write the first four terms of the sequence

Ex. Write an expression for an: b) 2, -5, 10, -17,...

A sequence is recursive if each term is defined by one or more previous terms Ex. The Fibonacci sequence is defined as a0 = 1, a1 = 1, ak = ak – 1 + ak – 2. Write the first six terms. Ex. Find the first five terms of the recursive sequence defined by a1 = 3, ak = 2ak – 1 – 5

If n is a positive integer, n factorial is defined as n! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ ... ∙ n As a special case, 0! = 1. Keep in mind that parentheses matter: 2n! = 2 ∙ n! = 2(1 ∙ 2 ∙ 3 ∙ 4 ∙ ... ∙ n) (2n)! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ ... ∙ 2n

Ex. Write the first five terms of the sequence

Ex. Evaluate the factorial b) c)

The Greek letter sigma (Σ) can be used to show the sum of many terms i is called the index of the summation n is the upper limit of the summation 1 is the lower limit of the summation

Ex. Find the sum a) b) c)

Consider the infinite sequence a1, a2, a3,..., ai,... The sum of the first n terms is called the nth partial sum of the sequence, and is denoted The sum of all the terms of the infinite sequence is called an infinite series, and is denoted

Ex. Use the first 3 partial sums to evaluate the sum

Practice Problems Section 8.1 Problems 1, 17, 37, 51, 59, 67, 73, 99

Arithmetic Sequences and Series A sequence is arithmetic if the difference of two consecutive terms is the same. an + 1 – an = d for any positive integer n The number d is called the common difference 15 15 15

Ex. Find the first 4 terms of the arithmetic sequence. a) an = 4n + 3 b) an = 7 – 5n c)

To find the nth term of an arithmetic sequence, we use the formula an = a1 + d(n – 1) where a1 is the first term and d is the common difference Ex. Find the nth term of the sequence 2, 5, 8, 11, 14,...

Ex. The fourth term of an arithmetic sequence is 20 and the 13th term is 65. Find the nth term.

Ex. Find the 9th term of the arithmetic sequence that starts with 2 and 9.

To find the sum of a finite arithmetic sequence with n terms, we use the formula Ex. Find the sum of the first 10 odd numbers. Ex. Find the sum of the integers from 1 to 100

Ex. Find the 150th partial sum of the arithmetic sequence 5, 16, 27, 38, 49,...

Ex. Find the sum

Ex. In a golf tournament, 16 golfers win cash prizes Ex. In a golf tournament, 16 golfers win cash prizes. First place gets $1000, second place gets $950, third place gets $900, and so on. What is the total amount of prize money?

Practice Problems Section 8.2 Problems 1, 21, 37, 45, 63, 65, 69, 89

Geometric Sequences and Series A sequence is geometric if the quotient of two consecutive terms is the same. for any positive integer n The number r is called the common ratio 25 25 25

Ex. Find the first 4 terms of the geometric sequence. a) an = 2n b) an = 4(3n) c) d) an = n2

To find the nth term of a geometric sequence, we use the formula an = a1rn – 1 where a1 is the first term and r is the common ratio Ex. Find the nth term of the sequence 3, 6, 12, 24, 48,...

Ex. Write the 15th term of the geometric sequence whose 1st term is 20 and whose common ratio is 1.05.

Ex. Write the 12th term of the geometric sequence 5, 15, 45,...

Ex. The 4th term of a geometric sequence is 125 and the 10th term is Ex. The 4th term of a geometric sequence is 125 and the 10th term is . Find the 14th term.

To find the sum of finite geometric sequence with n terms, we use the formula Ex. Find the sum

It is possible to take the sum of an infinite geometric sequence and get a finite answer. Consider We say that this geometric series converges A geometric series will converge if |r| < 1, and the sum is given by the formula

Ex. Find the sum a) b) 0.3 + 0.03 + 0.003 +...

Ex. A deposit of $50 is made on the first day of every month in an account that pays 6% compounded monthly. What is the balance at the end of 2 years?

Practice Problems Section 8.3 Problems 1, 11, 27, 35, 41, 57, 79, 107