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U SING AND W RITING S EQUENCES

U SING AND W RITING S EQUENCES. You can think of a sequence as a set of numbers written in a specific order. ( Any sequence can be defined as a function whose domain is the set of natural numbers .). The numbers (outputs) of a sequence are called terms. U SING AND W RITING S EQUENCES.

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U SING AND W RITING S EQUENCES

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  1. USING AND WRITING SEQUENCES You can think of a sequence as a set of numbers written in a specific order. (Any sequence can be defined as a function whose domain is the set of natural numbers.) The numbers (outputs) of a sequence are called terms.

  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 If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the nthterm of the sequence. Describe the pattern, write the next term, and write a rule for the n th 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. Examples from textbook: • Example 2 in text (p. 824) • Example 4 (p. 825) – define “recursive”sequence. • Define “partial sums” (p. 827) • Examples 5 and 6

  9. 5 ∑ 3i i=1 5 3 + 6 + 9 + 12 + 15 = ∑3i i = 1 SUMMATION Notation (aka SIGMA Notation) Is read as “the sum of 3i from iequals 1 to 5.” upper limit of summation index of summation lower limit of summation

  10. SUMMATION Notation (aka SIGMA Notation) 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 (but often does).

  11. Writing Series with Summation Notation . . . 5 + 10 + 15 + + 100 The summation notation is: Write this series using 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

  12. SOLUTION: Example: Write the series represented by the summation notation . Then find the sum.

  13. Writing Series with Summation Notation i ai= where i = 1, 2, 3, 4 . . . i + 1 The summation notation for the series is: Write the series using 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:

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