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U SING AND W RITING S EQUENCES

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. U SING AND W RITING S EQUENCES. n. a n.

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U SING AND W RITING S EQUENCES

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  1. 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.

  2. 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.

  3. 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

  4. 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

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

  6. Writing Rules for Sequences 5 n 1 243 1 3 1 9 1 27 1 81 - terms , , ,   1 2 3 4 5 rewrite terms 1 3 1 3 1 3 1 3 1 3 , , , - - - - - n 1 3 A rule for the nth term is an = - SOLUTION 12 3 4

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

  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 producedepartment of a grocery storeand 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 n 1 2 3 4 = 22 an 1 = 12 9 = 32 SOLUTION The diagram below shows the first three layers of the stack. Let an represent the number of oranges in layer n. From the diagram, you can see that an= n2

  11. Graphing a Sequence an= n2 Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100).

  12. 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

  13. 5  3i i= 1 5 3 + 6 + 9 + 12 + 15 = 3i i = 1 USING SERIES Is read as “the sum from iequals 1 to 5 of 3i.” upper limit of summation index of summation lower limit of summation

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

  15. USING SERIES 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 . . . 5 + 10 + 15 + + 100 20 5i. The summation notation is i = 1 Write the series with summation notation. SOLUTION 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 termsof the series can be written as: ai= 5i where i = 1, 2, 3, . . . , 20

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

  18. Writing Series with Summation Notation The sum of the terms of a finite sequence can be foundby simply adding the terms. For sequences with manyterms, 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 CONCEPT FORMULAS FOR SPECIAL SERIES SUMMARY n 1 = n i = 1 3 1 2 n n(n + 1) i = 2 i = 1 n n(n + 1)(2n + 1) i2 = 6 i = 1 gives the sum of n1’s. gives the sum of positive integers from 1 to n. gives the sum of squares of positive integers from 1 to n.

  20. Using a Formula for a Sum RETAIL DISPLAYSHow many oranges are in a square pyramid 10 layers high?

  21. Using a Formula for a Sum 10 . . .  i2 = 12+ 22 + + 102 i = 1 10(10 + 1)(2 • 10 + 1) = 6 10(11)(21) = 6 SOLUTION You know from the earlier example that the ith term of the series is given by ai= i2, where i = 1, 2, 3, . . . , 10. = 385 There are 385 oranges in the stack.

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