U SING AND W RITING S EQUENCES The numbers (outputs) of a sequence are called terms. sequence 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 domain gives the relative position of each term DOMAIN: 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

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

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

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 nth 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 Writing Rules for Sequences S OLUTION n terms    1, 1313  2, 1313  3, 1313  5 rewrite terms 1313  A rule for the nth term is: a n = n

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,… rewrite terms 1(1 +1)2(2 +1)3(3 +1)4(4 +1) n

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

= ∑ 3i 5 i = 1 SUMMATIONSIGMA SUMMATION Notation (aka SIGMA Notation) 5 i =1 ∑ 3i3i Is read as “the sum of 3 i from i equals 1 to 5.” index of summationlower limit of summation upper limit of summation

SUMMATIONSIGMA SUMMATION Notation (aka SIGMA Notation) i 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).

Writing Series with Summation Notation Write this series using summation notation: 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:

Example: Write the series represented by the summation notation. Then find the sum. S OLUTION:

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 using summation notation. S OLUTION: The summation notation for the series is: