HW# Do Now Aim : How do we write the terms of a Sequence ?

2 We will learn how to: SECTION use sequence notation …find specific and general terms in a sequence.

sequence A sequence is simply list of things generated by a rule.

In this chapter, we study sequences and series of numbers. list A sequence is a list of numbers (separated by commas). adds A series adds the numbers in the list together. Example: Sequence: 1, 2, 3, 4, …, n, … Series: …+ n + …

What symbol(s) do we use For a sequence? For a series?

Roughly speaking, a sequence is a list of numbers written in a specific order. The numbers in the sequence are often written as: a 1, a 2, a 3,.... The dots mean that the list continues forever. Here’s a simple example: 5, 10, 15, 20, 25,...

Sequences arise in many real-world situations.  If you deposit a sum of money into an interest-bearing account, the interest earned each month forms a sequence.  If you drop a ball and let it bounce, the height the ball reaches at each successive bounce is a sequence.

A sequence is a set of numbers written in a specific order: a 1, a 2, a 3, a 4, …, a n, … The number a 1 is called the first term, a 2 is the second term, and in general a n is the nth term. Since for every natural number n, there is a corresponding number a n, we can define a sequence as a function.

Example 1—Finding the Terms of a Sequence Find the first five terms and the 100th term of the sequence defined by each formula. To find the first five terms, we substitute n = 1, 2, 3, 4, and 5 in the formula for the nth term. To find the 100th term, we substitute n = 100.

Example 1—Finding the Terms of a Sequence This gives the following.

Model#1: Write the general term a n for a sequence whose first five terms are given. 2, 4, 6, 8, 10,…

Notice how the formula a n = 2n gives all the terms of the sequence. For instance, substituting 1, 2, 3, and 4 for n gives the first four terms: To find the 103rd term, we use n = 103 to get: a 103 = 2 · 103 = 206 a 1 = 2 · 1 = 2 a 2 = 2 · 2 = 4 a 3 = 2 · 3 = 6 a 4 = 2 · 4 = 8

13 EXAMPLE 2 Finding a General Term of a Sequence from a Pattern Write the general term a n for a sequence whose first five terms are given. Hint 1.Write the position number of the term above each term of the sequence 2.Look for a pattern that connects the term to the position number of the term.

14 EXAMPLE 3 Finding a General Term of a Sequence from a Pattern Solution a. Apparent pattern: Here 1 = 1 2, 4 = 2 2, 9 = 3 2, 16 = 4 2, and 25 = 5 2. Each term is the square of the position number of that term. This suggests a n = n 2.

15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Finding a General Term of a Sequence from a Pattern Solution continued b. Apparent pattern: When the terms alternate in sign and n = 1, we use factors such as (−1) n if we want to begin with the factor −1 or we use factors such as (−1) n+1 if we want to begin with the factor 1.

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Finding a General Term of a Sequence from a Pattern Solution continued b. continued Notice that each term can be written as a quotient with denominator equal to the position number and numerator equal to one less than the position number, suggesting the general term

Practice

Some sequences do not have simple defining formulas like those of the preceding example. The nth term of a sequence may depend on some or all of the terms preceding it. A sequence defined in this way is called recursive.

Model #2: Finding Terms of Recursively Defined Sequence Find the first five terms of the sequence defined recursively by a 1 = 1 and a n = 3(a n–1 + 2) The defining formula for the sequence is recursive. It allows us to find the nth term a n if we know the preceding term a n–1.

So, we can find the second term from the first term, the third term from the second term, the fourth term from the third term, and so on. Since we are given the first term a 1 = 1, we can proceed as follows. Model #2: Finding Terms of Recursively Defined Sequence

a 2 = 3(a 1 + 2) = 3(1 + 2) = 9 a 3 = 3(a 2 + 2) = 3(9 + 2) = 33 a 4 = 3(a 3 + 2) = 3(33 + 2) = 105 a 5 = 3(a 4 + 2) = 3( ) = 321 The first five terms of this sequence are: 1, 9, 33, 105, 321,… Model #2: Finding Terms of Recursively Defined Sequence

EXAMPLE 4 Finding Terms of a Recursively Defined Sequence Write the first five terms of the recursively defined sequence a 1 = 4, a n+1 = 2a n – 9 Solution We are given the first term of the sequence: a 1 = 4. a 2 = 2a 1 – 9 = 2(4) – 9 = –1 a 3 = 2a 2 – 9 = 2(–1) – 9 = –11 a 4 = 2a 3 – 9 = 2(–11) – 9 = –31 a 5 = 2a 4 – 9 = 2(–31) – 9 = –71 So the first five terms of the sequence are: 4, –1, −11, −31, −71

Regents Practice