8 1 Sequences Definition of a Sequence the

8. 1 Sequences

Definition of a Sequence □ the terms of the sequence (the subscript indicates the term’s “place in line”)

Explicit vs. Implicit A sequence can be defined implicitly and explicitly. Consider the sequence 1, 3, 6, 10, 15, …

Finding Terms of a Sequence □ The first four terms of the sequence given by are

Finding Terms of a Sequence □ Solution: The first five terms of the sequence are 2, 4, 10, 28, 82.

Have you ever seen this sequence before? □ 1, 1, 2, 3, 5, 8 … □ Can you find the next three terms in the sequence? □ Hint: 13, □ 21, 34 □ Can you explain this pattern?

The Fibonacci Sequence □ The Fibonacci Sequence is defined as follows: Write the first six terms of the Fibonacci Sequence:

Finding the nth term of a Sequence □ Write an expression for the apparent nth term (an) of each sequence. □ a. 1, 3, 5, 7, … b. 2, 5, 10, 17, … Solution: a. n: 1 2 3 4. . . n terms: 1 3 5 7. . . an Apparent pattern: Each term is 1 less than twice n, which implies that b. n: 1 2 3 4 … n terms: 2 5 10 17 … an Apparent pattern: Each term is 1 more than the square of n, which implies that

Additional Example □ Write an expression for the apparent nth term of the sequence: Solution: Apparent pattern: Each term has a numerator that is 1 greater than its denominator, which implies that

Factorial Notation □ If n is a positive integer, n factorial is defined by As a special case, zero factorial is defined as 0! = 1. Here are some values of n! for the first several nonnegative integers. Notice that 0! is 1 by definition. The value of n does not have to be very large before the value of n! becomes huge. For instance, 10! = 3, 628, 800.

Finding the Terms of a Sequence Involving Factorials □ List the first five terms of the sequence given by Begin with n = 0.

Arithmetic Sequences □

Example Write an explicit AND implicit formula for the sequence: 7, 10, 13, 16, …

Geometric Sequences □

Example Write an explicit AND implicit formula for the sequence: 125, 5, 1, …

Example If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term.

Does converge? The sequence converges and its limit is 2.

Absolute Value Theorem for Sequences If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero. p