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Sequences

Sequences. What is a sequence?. A sequence of numbers is a list, an ordering that may or may not follow some rule. Arithmetic Sequence : Geometric Sequence :. What are two types of Sequences?.

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Sequences

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  1. Sequences What is a sequence? A sequence of numbers is a list, an ordering that may or may not follow some rule. • Arithmetic Sequence: • Geometric Sequence: What are two types of Sequences? Is a sequence in which each term is equal to the previous term plus a constant. This constant is called the common difference. Is a sequence in which each term is equal to the previous term multiplied by a constant. This constant is called the common ratio.

  2. When we are given a sequence and asked to find a specific term… 1.Determine whether the sequence is arithmetic or geometric. 2. Find the common difference (arithmetic) or the common ratio (geometric) 3. Use the appropriate formula to find the term asked for. Arithmetic Geometric

  3. n =10 because we are looking for the 10th term d = -6 because we are adding -6 to each term = 12 because it is the first term of the sequence We will be reviewing arithmetic Find the common difference, find the indicated term, write the equation in both function and explicit form Let’s look at a sequence. 12, 6, 0, -6, . . . Find Write down every thing you need in the formula.

  4. Now use the information in the formula…don’t plug in “n” and you will have the explicit form This is the explicit form Distribute the -6 and simplify, change to f(n) and you will have function form This is function form To find the 10th term use n = 10 and simplify

  5. So let’s list some steps • First find the common difference • Next list all of the unknowns • Find the explicit form by plugging in the first term and the common difference • Find the function form by simplifying the explicit form • To find the specific term replace n with the term you are looking for.

  6. You try… Find the common difference, the indicated term, write the explicit and function form of the sequence. • 6,106, 206, 306,…n=15 • -38,-45,-52,-59,…n=23 • -16,14, 44, 74,…n=52 1406 -206 1574

  7. What is a recursive sequence? Definition: A recursive sequence is the process in which each step of a pattern is dependent on the step or steps before it. Recursion Formulas: A recursion formula defines the nth term of a sequence as a function of the previous term. If the first term of a sequence is known, then the recursion formula can be used to determine the remaining terms.

  8. Sequence and Terms n² Let’s look at the following sequence Do you know what the rule is for the sequence? 1, 4, 9, 16, 25, 36, 49, …, The letter a with a subscript is used to represent function values of a sequence. The subscripts identify the location of a term.

  9. How to read the subscripts: a term in the sequence the prior term the next term

  10. Example 1: Find the first four terms of the sequence: General Term Let’s be sure we understand what is given + 2 is Each term after the first 3 times the previous term Plus 2 The first term is 5

  11. Continued…EX 1: Find the first four terms of the sequence: Start with general term for n>1 n=1 given n=2 n=3 n=4 Answer = 5, 17, 53, 161

  12. Your turn:Ex 2: Find the next four terms of the sequence. Start with general term for n>1 n=1 given n=2 n=3 n=4 Answer = 3, 6, 12, 24

  13. Write a recursive formula for the arithmetic sequence below.Step 1 : Make sure it is arithmeticStep 2 : Plug into the arithmetic recursive formula.Step 3 : Make sure you tell us what a1 is equal to. Ex. 3 7, 3, -1, -5, -9, … The common difference = -4 The first term = 7

  14. Last Example  Choose the recursive formula for the given sequence. Answer = C

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