Transcript Lesson 34
Bell Quiz Objectives • Determine whether or not a sequence is arithmetic. • Write a recursive formula for an arithmetic sequence. • Find the nth term of an arithmetic sequence Sequence • Sequences of numbers can be formed using a variety of patterns and operations. • A sequence is a list of numbers that follow a rule – each number in the sequence is called a term of the sequence. • Here are a few examples of sequences: • • • • 1, 3, 5, 7, … 7, 4, 1, –2, … 2, 6, 18, 54, … 1, 4, 9, 16, … Arithmetic Sequence • In the previous examples, the first two sequences are a special type of sequence called an arithmetic sequences. • An arithmetic sequence is a sequence that has a constant difference between two consecutive terms called the common difference. Arithmetic Sequence • To find the common difference, choose any term and subtract the previous term. • In the first sequence, the common difference is 2. • In the second sequence, the common difference is –3 Example 1 Recognizing Arithmetic Sequences Determine if the sequence is an arithmetic sequence. If yes, find the common difference and the next two terms. 7, 12, 17, 22, … Example 2 Recognizing Arithmetic Sequences Determine if the sequence is an arithmetic sequence. If yes, find the common difference and the next two terms. 3, 6, 12, 24, … Lesson Practice Determine if the sequence is an arithmetic sequence. If yes, find the common difference and the next two terms. 7, 6, 5, 4, … Lesson Practice Determine if the sequence is an arithmetic sequence. If yes, find the common difference and the next two terms. 10, 12, 15, 19, … Arithmetic Sequence • The first term of a sequence is denoted as 𝑎1 , the second term as 𝑎2 , the third term 𝑎3 , and so on. • The nth term of an arithmetic sequence is denoted as 𝑎𝑛 . • The term preceding 𝑎𝑛 is denoted 𝑎𝑛 − 1 . • For example, if n = 6 then the term preceding 𝑎6 is 𝑎6 − 1 or 𝑎5 . Arithmetic Sequence Arithmetic Sequence • Arithmetic sequences can be represented using a formula Example 3 Using a Recursive Formula Use a recursive formula to find the first four terms of an arithmetic sequence where 𝑎1 = – 2 and the common difference d = 7. Lesson Practice Use a recursive formula to find the first four terms of an arithmetic sequence where 𝑎1 = – 3 and the common difference d = 4. Arithmetic Sequence • A rule for finding any term in an arithmetic sequence can be developed by looking at a different pattern in the sequence 7, 11, 15, 19, … Arithmetic Sequence • To find the nth term of an arithmetic sequence we can use the formula: Example 4 Finding the nth Term in Arithmetic Sequences Use the rule 𝑎𝑛 = 6 + (n – 1)2 to find the 4th and 11th terms of the sequence. Lesson Practice Use the rule 𝑎𝑛 = 14 + (n – 1)(– 3) to find the 4th and 11th terms of the sequence. Example 5 Finding the nth Term in Arithmetic Sequences Find the 10th term of the sequences 3, 11, 19, 27, … Lesson Practice Find the 10th term of the sequences 1, 10, 19, 28, … Example 6 Finding the nth Term in Arithmetic Sequences Find the 10th term of the sequences 1 3 5 7 , , , , 4 4 4 4 … Lesson Practice Find the 11th term of the sequences 2 , 3 1, 1 1 , 3 1 2 , 3 … Example 7 Application: Seating for a Reception The first table at a reception will seat 9 guest while each additional table will seat 6 more guests. a. Write a rule to model the situation. b. Use the rule to find how many guests can be seated with ten tables. Lesson Practice Flowers are purchased to put on tables at a reception. The head table needs to have 12 flowers and the other tables need to have 6 flowers each a. Write a rule to model the situation. b. Use the rule to find the number of flowers needed for 15 tables.