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Arithmetic Sequences

Arithmetic Sequences. How do I define an arithmetic sequence and how do I use the formula to find different terms of the sequence?. U SING AND W RITING S EQUENCES. The numbers in sequences are called terms.

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Arithmetic Sequences

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  1. Arithmetic Sequences How do I define an arithmetic sequence and how do I use the formula to find different terms of the sequence?

  2. USING AND WRITING SEQUENCES The numbers in sequences are called terms. You can think of a sequence as a function whose domainis a set of consecutive integers. If a domain is notspecified, it is understood that the domain starts with 1.

  3. USING AND WRITING SEQUENCES n an 1 2 3 4 5 DOMAIN: The domain gives the relative positionof each term. The range gives the terms of the sequence. 3 6 9 12 15 RANGE: This is a finite sequence having the rule an= 3n, where anrepresents the nth term of the sequence.

  4. Writing Terms of Sequences Write the first six terms of the sequence an = 2n + 3. SOLUTION a1= 2(1) + 3 = 5 1st term a2= 2(2) + 3 = 7 2nd term a3= 2(3) + 3 = 9 3rd term a4= 2(4) + 3 = 11 4th term a5= 2(5) + 3 = 13 5th term a6= 2(6) + 3 = 15 6th term

  5. Writing Terms of Sequences Write the first six terms of the sequence f(n) = (–2)n – 1 . SOLUTION f(1) = (–2)1 – 1 = 1 1st term f(2) = (–2)2 – 1 = –2 2nd term f(3) = (–2)3 – 1 = 4 3rd term f(4) = (–2)4 – 1 = – 8 4th term f(5) = (–2)5 – 1 = 16 5th term f(6) = (–2)6 – 1 = – 32 6th term

  6. Arithmetic Sequences and Series Arithmetic Sequence: sequence whose consecutive terms have a common difference. Example:3, 5, 7, 9, 11, 13, ... The terms have a common difference of 2. The common difference is the number d. To find the common difference you use an+1 – an Example: Is the sequence arithmetic? –45, –30, –15, 0, 15, 30 Yes, the common difference is 15 How do you find any term in this sequence? To find any term in an arithmetic sequence, use the formula an = a1 + (n – 1)d where d is the common difference.

  7. The first term of an arithmetic sequence is . We add d to get the next term. There is a pattern, therefore there is a formula we can use to give use any term that we need without listing the whole sequence . The nth term of an arithmetic sequence is given by: The last # in the sequence/or the # you are looking for The position the term is in First term The common difference

  8. Examples: Find the 14th term of the arithmetic sequence 4, 7, 10, 13,……

  9. Examples: In the arithmetic sequence 4,7,10,13,…, which term has a value of 301?

  10. Vocabulary of Sequences (Universal)

  11. Given an arithmetic sequence with x 38 15 NA -3 X = 80

  12. -6 20 29 NA x

  13. 9 633 x NA 24 X = 27

  14. 1.5 x 16 NA 0.5 Try this one:

  15. Example: Find a formula for the nth term of the arithmetic sequence in which the common difference is 5 and the first term is 3. an = a1 + (n – 1)d a1 = 3 d = 5 an = 3 + (n – 1)5

  16. Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence? an = a1 + (n – 1)d We need to determine what the first term is... d = 4 and a5 = 15 a5 = a1 + (5 – 1)4 = 15 a1 = –1 a10= –1 + (10 – 1)4 a10 = 35

  17. An arithmetic meanof two numbers, a and b, is simply their average. Using the arithmetic mean we can also form a sequence. Examples: Insert three arithmetic means between 8 and 16. Let 8 be the 1st term Let 16 be the 5th term 10 12 14

  18. The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence Find two arithmetic means between –4 and 5 -4, ____, ____, 5 -4 5 4 NA x

  19. The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 4 5 NA x

  20. FINITE SEQUENCE INFINITE SEQUENCE 3, 6, 9, 12, 15 3, 6, 9, 12, 15, . . . FINITE SERIES INFINITE SERIES 3 + 6 + 9 + 12 + 15 3 + 6 + 9 + 12 + 15 + . . . 5 3 + 6 + 9 + 12 + 15 = ∑3i i = 1 USING SERIES When the terms of a sequence are added, the resultingexpression is a series. A series can be finite or infinite. . . . You can use summation notation to write a series. Forexample, for the finite series shown above, you can write

  21. UPPER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE) LOWER BOUND (NUMBER)

  22. An arithmetic seriesis a series associated with an arithmetic sequence. The sum of the first n terms: Definition:

  23. Examples: Find the sum of the first 100 natural numbers. 1 + 2 + 3 + 4 + … + 100

  24. Find the sum of the first 14 terms of the arithmetic series 2 + 5 + 8 + 11 + 14 + 17 +… Examples:

  25. Examples: Find the sum of the series

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