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. U SING AND W RITING S EQUENCES The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.

3. The domain gives the relative position of each term. 1 2 3 4 5 DOMAIN: 3 6 9 12 15 RANGE: The range gives the terms of the sequence. This is a finite sequence having the rule a n = 3n, where a n represents the n th term of the sequence. U SING AND W RITING S EQUENCES n anan

4. Writing Terms of Sequences Write the first six terms of the sequence a n = 2n + 3. S OLUTION a 1 = 2(1) + 3 = 5 1st term 2nd term 3rd term 4th term 6th term a 2 = 2(2) + 3 = 7 a 3 = 2(3) + 3 = 9 a 4 = 2(4) + 3 = 11 a 5 = 2(5) + 3 = 13 a 6 = 2(6) + 3 = 15 5th term

5. Writing Terms of Sequences Write the first six terms of the sequence f (n) = (–2) n – 1. S OLUTION f (1) = (–2) 1 – 1 = 1 1st term 2nd term 3rd term 4th term 6th term f (2) = (–2) 2 – 1 = –2 f (3) = (–2) 3 – 1 = 4 f (4) = (–2) 4 – 1 = – 8 f (5) = (–2) 5 – 1 = 16 f (6) = (–2) 6 – 1 = – 32 5th 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 a n+1 – a n 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 formulaa n = a 1 + (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 First term The position the term is in The common difference

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

9. 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 15 38 NA -3 X = 80

12. -6 29 20 NA x

13. 9 x 633 NA 24 X = 27

14. Try this one: 1.5 16 x NA 0.5

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. a n = a 1 + (n – 1)d a 1 = 3 d = 5 a n = 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? a n = a 1 + (n – 1)d We need to determine what the first term is... d = 4 and a 5 = 15 a 5 = a 1 + (5 – 1)4 = 15 a 1 = –1 a 10 = –1 + (10 – 1)4 a 10 = 35

17. An arithmetic mean of two numbers, a and b, is simply their average. Using the arithmetic mean we can also form a sequence. Insert three arithmetic means between 8 and 16. Let 8 be the 1 st term Let 16 be the 5 th term 101214

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

19. Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 5 4 NA x 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

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

21. An arithmetic series is a series associated with an arithmetic sequence. The sum of the first n terms:

22. Find the sum of the first 100 natural numbers. 1 + 2 + 3 + 4 + … + 100

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