A sequence is an ordered list of numbers. Each number in a sequence is called a term. a 1, a 2, a 3,..., a n

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1 Algebra 2/Trig Unit 8 Sequences and Series Lesson 1 I can identify a pattern found in a sequence. I can use a formula to find the nth term of a sequence. I can write a recursive formula for a sequence. I can write an explicit formula for a sequence. I can find, identify, and apply arithmetic and geometric sequences. Vocabulary: Terms: The numbers in the sequence Infinite Sequence: A sequence that continues without end Recursive Formula: gives a rule or the nth term a n of the sequence as a function of the preceding term a n-1. Explicit Formula: gives a ule for the nth term a n of the sequence as a function of the position n of the term within the sequence. Sequence: A function whose domain is a set of consecutive integers. Finite Sequence: A sequence that has a last term. Arithmetic Sequence: A sequence in which the difference between consecutive terms is constant. Common Difference: The constant difference between consecutive terms of an arithmetic sequence. Geometric Sequence: A sequence in which the ration of any term to the previous term is constant. Common Ratio: The constant ratio between consecutive terms of a geometric sequence. A sequence is an ordered list of numbers. Each number in a sequence is called a term. a 1, a 2, a 3,..., a n Explicit Formula: describes the n th term of a sequence using the number n. Do NOT need to know previous term. Example: 2, 4, 6, 8, 10 Explicit Formula is a n = 2n Given the explicit Formula: a n =3n 2. What are first 3 terms of the sequence? a 1 =3( a 2 =3( a 3 =3( ) - 2 = ) - 2 = ) - 2 = 3n Explicit Practice: Given an Find the 8 th term of the sequence. n 2 *You do not need to know the previous value in order to determine the n th value.

2 Recursive Formula: relates each term after the first term to the one before it.

Recursion is the process of choosing a starting term and

4 Recursion is the process of choosing a starting term and repeatedly applying the same process to each term to arrive at the following term. Recursion requires that you know the value of the term immediately before the term you are trying to find. A recursive formula always has two parts: 1. The starting value for a the recursion equation for a n as a function of a n-1 (the term before it.)

5 Recursive Practice: Given a n = 3a n-1 1 find the next three terms of the sequence. a 1 = 4 Some notes that follow are from the District Regents Prep Web site: Created by Donna Roberts Copyright Examples: Arithmetic Sequence Common Difference, d 1, 4, 7, 10, 13, 16,... 15, 10, 5, 0, -5, -10,... 1, ½, 0, - ½,... Important arithmetic sequence formula: To find any term of an arithmetic sequence: a n = a 1 + (n 1)d a 1 is first term of the sequence, d is the common difference, n, is the number of the term to find a 1 is often simply referred to as a

6 9. Insert 3 arithmetic means between 7 and 23. 7,,,, 23,...

Important formula used with geometric sequences: a n = a 1 r n-1 where a 1 is the first term of the sequence, r is the common ratio, and n is the number of the term to find. 1. Find the common ration for the sequence 3 3 6, 3,,,.

7 Important formula used with geometric sequences: a n = a 1 r n-1 where a 1 is the first term of the sequence, r is the common ratio, and n is the number of the term to find. 1. Find the common ration for the sequence 3 3 6, 3,,, Find the common ratio for the sequence given the formula a n = 5(3) n-1 3. Find the 7 th term of the sequence 2, 6, 18, 54, Find the 11 th term of the sequence ,,,, Find a 8 for the sequence 0.5, 3.5, 24.5, 171.5, The third term of a geometric sequence is 3 and the sixth term is 1. Find the first term. 9 Now see hand out for rest of practice problems.

8 How to do problems on calculator.

12 The following will be distributed in class to all students. Name: Period: Time for you to try some practice problems. You can show work here or on separate paper. 1. Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, Find the 9 th term of the sequence 1, 2,2, Insert three geometric means between 1 and 81. 1,,,, Find a 6 for an arithmetic sequence where a 1 = 3x + 1 and d = 2x Find t 12 for a geometric sequence where t 1 = 2 + 2i and r = 3. Word Problems 1. You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence. What is the total distance the object will fall in 6 seconds? 2. The sum of the interior angles of a triangle is 180 o, of a quadrilateral is 360 o, and of a pentagon is 540 o. Assuming this pattern continues, find the sum of the interior angles of a dodecagon (12 sides). 3. After knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase that time by 6 inutes per day. How many weeks will it be before you are up to jogging 60 minutes per day?

13 4. You complain that the hot tub in your hotel suite I not hot enough. The hotel tells you that they will increase the temperature by 10% each hour. If the current temperature of the hot tub is 75 o F, what will be the temperature of the hot tub after 3 hours, to the nearest tenth of a degree? 5. A culture of bacteria doubles every 2 hours. If there are 500 bacteria at the beginning, how many bacteria will there be after 24 hours?

Lesson 4. Exit Ticket Sample Solutions. explicit formula. b. for and

Lesson 4. Exit Ticket Sample Solutions. explicit formula. b. for and Exit icket Sample Solutions. Write the first three terms in the following geometric sequences. hen write the explicit formula. a. for and,, b. for and,, or,,.. Write an explicit formula for the geometric