1. 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.

2. 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

3. 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

4. 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

5. Writing Rules for Sequences If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the n th term of the sequence. Describe the pattern, write the next term, and write a rule for the n th term of the sequence __ –,, –,, …. 1 3 1 9 1 27 1 81 __

6. 1 3 , 1 9, 1 27 , 1 81 Writing Rules for Sequences S OLUTION 1 2 3 4 n terms 1 243  5 1313  4 1313  1, 1313  2, 1313  3, 1313  5 rewrite terms 1313  A rule for the n th term is a n = n

7. 2 6 12 20 Writing Rules for Sequences S OLUTION A rule for the n th term is f (n) = n (n+1). terms 5(5 +1) Describe the pattern, write the next term, and write a rule for the n th term of the sequence. 2, 6, 12, 20,…. 5 30 1 2 3 4 rewrite terms 1(1 +1)2(2 +1)3(3 +1)4(4 +1) n

8. Graphing a Sequence You can graph a sequence by letting the horizontal axis represent the position numbers (the domain) and the vertical axis represent the terms (the range).

9. Graphing a Sequence You work in the produce department of a grocery store and are stacking oranges in the shape of square pyramid with ten layers. Write a rule for the number of oranges in each layer. Graph the sequence.

10. Graphing a Sequence S OLUTION The diagram below shows the first three layers of the stack. Let a n represent the number of oranges in layer n. n123 anan 1 = 1 2 4 = 2 2 9 = 3 2 From the diagram, you can see that a n = n 2

11. Graphing a Sequence Plot the points (1, 1), (2, 4), (3, 9),..., (10, 100). a n = n 2

12. 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.

13. 3 + 6 + 9 + 12 + 15 =  3i 5 i = 1 U SING S ERIES 5 i = 1  3i3i Is read as “the sum from i equals 1 to 5 of 3i.” index of summationlower limit of summation upper limit of summation

14. U SING S ERIES Summation notation is also called sigma notation because it uses the uppercase Greek letter sigma, written . Summation notation for an infinite series is similar to that for a finite series. For example, for the infinite series shown earlier, you can write: 3 + 6 + 9 + 12 + 15 + =  3i  i = 1... The infinity symbol, , indicates that the series continues without end.

15. U SING S ERIES The index of summation does not have to be i. Any letter can be used. Also, the index does not have to begin at 1.

16. Writing Series with Summation Notation Write the series with summation notation. 5 + 10 + 15 + + 100... S OLUTION Notice that the first term is 5 (1), the second is 5 (2), the third is 5 (3), and the last is 5 (20). So the terms of the series can be written as: a i = 5i where i = 1, 2, 3,..., 20 The summation notation is  5i. 20 i = 1

17. Writing Series with Summation Notation Notice that for each term the denominator of the fraction is 1 more than the numerator. So, the terms of the series can be written as: a i = where i = 1, 2, 3, 4... i i + 1 Write the series with summation notation. S OLUTION... 1 2 3 4 2 3 4 5 + +  The summation notation for the series is  i = 1 i i + 1.

18. Writing Series with Summation Notation The sum of the terms of a finite sequence can be found by simply adding the terms. For sequences with many terms, however, adding the terms can be tedious. Formulas for finding the sum of the terms of three special types of sequences are shown next.

19. Writing Series with Summation Notation FORMULAS FOR SPECIAL SERIES C ONCEPT S UMMARY n i = 1  1 = n  i = n (n + 1) 2 n i = 1 1 2 3 gives the sum of positive integers from 1 to n. gives the sum of squares of positive integers from 1 to n.  i 2 = n (n + 1)(2 n + 1) 6 n i = 1 gives the sum of n 1’s.

20. Using a Formula for a Sum R ETAIL D ISPLAYS How many oranges are in a square pyramid 10 layers high?

21. Using a Formula for a Sum S OLUTION You know from the earlier example that the i th term of the series is given by a i = i 2, where i = 1, 2, 3,..., 10. 10  i = 1 i 2 = 1 2 + 2 2 + + 10 2... 10(11)(21) = 6 = 385 There are 385 oranges in the stack. = 6 10(10 + 1)(2 10 + 1)