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Algebra II – Sequences and Series

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    Study Guide Algebra II – Sequences and Series 1. Definition and Examples of Sequences A sequence is an ordered list of numbers that follows a certain pattern. Examples of sequences include: The three dots (…) show that the pattern continues. Each number in a sequence is called a term . For example, in the sequence: 1, 3, 5, 7, 9, … • 1 is the first term • 3 is the second term • 5 is the third term Terms are usually written using notation like: Here: • (a 1 ) = first term • (a 2 ) = second term • (a n ) = nth term The expression for (a n ) is called the general term or nth - term formula . It allows us to find any term in the sequence .

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    Study Guide Example 1: Finding Terms from a Formula Suppose the general term is: We want to find the first five terms . First term Second term Third term Fourth term Fifth term Result The first five terms are: 5, 8, 11, 14, 17 Example 2: Another Sequence Formula Now consider the formula: Find the first five terms. First term

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    Study Guide Second term Third term Fourth term Fifth term Result The first five terms are: 2, 6, 18, 54, 162 Example 3: Finding the Formula for a Sequence Sometimes we are given a sequence and must find the formula for the nth term . Sequence 1 2, 4, 6, 8, … Notice that each term is twice the term number . Formula

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    Study Guide Sequence 2 10, 50, 250, 1250, … Each term is 5 times the previous term , so this is a geometric sequence . Rewrite the numbers: Formula Sequence 3 3, 7, 11, 15, 19, … The pattern increases by 4 each time , so it is an arithmetic sequence . Write each term in terms of (n): Formula

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    Study Guide Simplify: Key Ideas to Remember • A sequence is an ordered list of numbers that follows a pattern. • Each number in the sequence is called a term . • The nth - term formula (a n ) allows us to find any term in the sequence. • Some sequences increase by a constant amount ( arithmetic sequences ). • Others multiply by a constant number ( geometric sequences ). Summary Sequences describe number patterns that follow specific rules. • The notation (a 1 , a 2 , a 3 , … ) represents the terms of a sequence. • The formula (a n ) gives the general rule for generating the sequence. • Arithmetic sequences grow by adding a constant difference . • Geometric sequences grow by multiplying by a constant ratio . 2. Quiz: Definition and Examples of Sequences 1. Question Find the next term in the sequence:

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    Study Guide Answer Choices • 6 • 7 • 8 Correct Answer : 7 Why This Is Correct Look at the difference between terms: So the sequence increases by 8 each time . Next term: So the next term is 7 . 2. Question Find the 7th term of the sequence described by: Answer Choices • 27 • 31 • 35 Correct Answer : 27

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    Study Guide Why This Is Correct Substitute (n = 7): So the 7th term is 27 . 3. Question Find the 9th term of the sequence described by: Answer Choices • 64 • 32 • 16 Correct Answer : 16 Why This Is Correct Substitute (n = 9): So the 9th term is 16 . 4. Question Find the 5th term of the sequence described by:

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    Study Guide Answer Choices • 91 • 213 • 253 Correct Answer : 253 Why This Is Correct Substitute (n = 5): So the 5th term is 253 . 5. Question Find the 11th term of the sequence described by: Answer Choices • 130 • 154 • 180 Correct Answer : 154 Why This Is Correct

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    Study Guide Substitute (n = 11): So the 11th term is 154 . 3. Arithmetic Sequence An arithmetic sequence is a sequence of numbers where each term after the first is found by adding the same number to the previous term . This constant number is called the common difference . General Formula The formula for the (n)th term of an arithmetic sequence is: Where: • (a n ) = the (n)th term • (a 1 ) = the first term • (d) = common difference • (n) = term number What is the Common Difference? The common difference is the amount added to each term to get the next term. It can be found using: This means you subtract the first term from the second term.

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    Study Guide Example 1 Find the common difference , write the formula for the sequence, and determine the 20th term . Sequence 1 1, 5, 9, 13, 17, … Step 1: Find the Common Difference So the common difference is 4 . Step 2: Write the Formula Using the formula Substitute the values: Simplify: Step 3: Find the 20th Term So the 20th term is 77 .

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