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# Arithmetic Sequences

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Recursive and Explicit Forms of Arithmetic Sequences

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### Arithmetic Sequences

1. 1. Arithmetic Sequences
2. 2. Section 1.3 Arithmetic Sequences An arithmetic sequence is a sequence in which the difference between each term and the preceding term is always constant. Which of the following sequences is arithmetic? a. {14, 10, 6, 2, -2, -6, -10, . . . } b. {3, 5, 8, 12, 17, . . . } a. yes, the difference between each term is -4 b. no, the difference between the first two terms is 2 and the difference between the 2nd and 3rd term is 3.
3. 3. Recursive Form of an Arithmetic Sequence Un = Un-1 + d for some constant d and all n > 2 The number d is called the common difference of the arithmetic sequence.
4. 4. Graph of an Arithmetic Sequence • If {Un} is an arithmetic sequence with U1 = 3 and U2 = 4.5 as its first two terms, • a. Find the common difference. • b. Write the sequence as a recursive function. • c. Give the first six terms of the sequence. • d. Graph the sequence.
5. 5. • a. Find the common difference. • U - U = 4.5 - 3 = 1.5 2 1 • The common difference is 1.5
6. 6. • b. Write the sequence as a recursive function. •U 1 = 3, Un = Un-1 + 1.5, for n > 2 First Method for Always one Term finding nth term greater than by using the subscript of first preceding term. term.
7. 7. • c. Give the first six terms of the sequence.
8. 8. Explicit Form of an Arithmetic Sequence • In an arithmetic sequence {Un} with common difference d, Un = U1 + (n - 1)d for every n > 1 Find the nth term of an arithmetic sequence with first term -5 and common difference of 3. Sketch a graph of the sequence. Un = U1 + (n - 1)d = -5 + (n - 1)3 = -5 + 3n - 3 = 3n - 8
9. 9. Find the nth term of an arithmetic sequence with first term -5 and common difference of 3. Sketch a graph of the sequence.
10. 10. Finding a Term of an Arithmetic Sequence What is the 45th term of the arithmetic sequence whose first three terms are 5, 9, and 13? First find d; d = 9 - 5 = 4 Second find explicit form: Un = 5 + (n - 1)4 Un = 4n + 1 Then find 45th term: U45 = 4(45) + 1 U45 = 181
11. 11. Finding Explicit and Recursive Formulas If {Un} is an arithmetic sequence with U6 = 57 and U10 = 93, find U1, a recursive formula, and an explicit formula for Un. To find d when given to non-consecutive terms use the formula: d = Um --Un m n d = 93 -- 57 10 6 =9
12. 12. Finding U1 Select either of the given terms and substitute into Explicit formula. Un = U1 + (n - 1)d U6 = 57 U10 = 93 57 = U1 + (6 - 1)9 93 = U1 + (10 - 1)9 U1 = 12 U1 = 12
13. 13. FORMULAS Explicit Form Recursive Form Un = U1 + (n - 1)d Un = Un-1 + d Un = 12 + (n - 1)9 Un = Un-1 + 9, for n > 2 Un = 9n + 3, for n > 1