Defining and Using Sequences and Series Section 8.1 beginning on page 410

Sequences A sequence is an ordered list of numbers. A finite sequence has an end and its domain is the set {1,2,3,…,𝑛}. The values in the range are called the terms of the sequence. An infinite sequence is a function that continues without stopping and whose domain is the set of positive integers. Finite sequence: 2,4,6,8 Infinite sequence: 2,4,6,8,… A sequence can be specified by an equation, or a rule. For example, both sequences above can be described by the rule 𝑎 𝑛 =2𝑛 or 𝑓 𝑛 =2𝑛

Writing the Terms of a Sequence Note: The domain of a sequence could begin with 0 instead of 1. Unless otherwise indicated, assume the domain of a sequence begins with 1. Example 1: Write the first six terms of (a) and (b). a) 𝑎 𝑛 =2𝑛+5 b) 𝑓 𝑛 = (−3) 𝑛−1 𝑎 1 = 2(1)+5 =7 First Term 𝑓(1)= (−3) 1 −1 =1 (−3) 2 −1 =−3 𝑎 2 = 2(2)+5 =9 Second Term 𝑓(2)= =9 𝑎 3 = 2(3)+5 =11 Third Term 𝑓(3)= (−3) 3 −1 𝑎 4 = 2(4)+5 =13 Fourth Term 𝑓(4)= (−3) 4 −1 =−27 𝑎 5 = 2(5)+5 =15 Fifth Term 𝑓(5)= (−3) 5 −1 =81 𝑎 6 = 2(6)+5 =17 Sixth Term 𝑓(6)= (−3) 6 −1 =−243

Writing Rules for Sequences Example 2a: Describe the pattern, write the next term, and write a rule for the nth term of the sequence. 𝑛=2 𝑛=3 𝑛=4 𝑛=1 −1,−8,−27,−64,… How are these terms related to each other? Can they be defined in a similar way? The terms are all perfect cubes. The next term would be (−5) 3 =−125 To write a rule, relate the terms to their relative position (the n values). (−1) 3 , (−2) 3 , −3 3 , (−4) 3 𝑎 𝑛 = (−𝑛) 3

Writing Rules for Sequences Example 2b: Describe the pattern, write the next term, and write a rule for the nth term of the sequence. 𝑛=2 𝑛=3 𝑛=4 𝑛=1 0, 2, 6, 12,… How are these terms related to each other? Can they be defined in a similar way? Can the terms can be re-written using their relative positions (n values) ? The next term would be 4(5)=20 To write a rule, relate the terms to their relative position (n values). 0(1),1(2),2(3),3(4) 𝑎 𝑛 = (𝑛−1)(𝑛)

Solving a Real-Life Problem ** Do not copy, just pay attention.

Writing Rules for Series When the terms of a sequence are added together, the resulting expression is a series. A series can be finite or infinite. Finite Series: 2+4+6+8 Infinite Series: 2+4+6+8+… 𝑖=1 4 2𝑖 Summation Notation/Sigma Notation: 𝑖=1 ∞ 2𝑖 Summation Notation/Sigma Notation: 𝑖 is the index (like the relative positon/n-value) and the lower limit of the summation. 4 is the upper limit in the finite series, and ∞ is the upper limit in the infinite series.

Writing Series Using Summation Notation Example 4: Write each series using summation notation. a) 25+50+75+⋯+250 b) 1 2 + 2 3 + 3 4 + 4 5 +⋯ ** try to re-write the terms using their index (relative position) 1 1+1 + 2 2+1 + 3 3+1 + 4 4+1 +⋯ 25 1 +25 2 +25 3 +⋯25(10) Ending point 𝑖=1 10 25𝑖 Ending point 𝑖=1 ∞ 𝑖 𝑖+1 Starting point Starting point

Finding the Sum of a Series Note: The index of a summation does not have to be 𝑖, it could be any variable, and it does not have to start at 1. Example 5: Find the sum 𝑘=4 8 (3+ 𝑘 2 ) =(3+ 4 2 ) +(3+ 5 2 ) +(3+ 6 2 ) +(3+ 7 2 ) +(3+ 8 2 ) =19+28+39+52+67 For series with many terms, finding the sum by adding the terms can be time consuming. There are formulas for special types of series. =205

Formulas For Special Series 𝑖 𝑛 1 =𝑛 Sum of n terms of 1: Sum of first n positive integers: Sum of squares of first n positive integers: 𝑖 𝑛 𝑖 = 𝑛(𝑛+1) 2 𝑖 𝑛 𝑖 2 = 𝑛(𝑛+1)(2𝑛+1) 6

Using a Formula For a Sum 𝑖 𝑛 1 =𝑛 𝑖 𝑛 𝑖 = 𝑛(𝑛+1) 2 𝑖 𝑛 𝑖 2 = 𝑛(𝑛+1)(2𝑛+1) 6 Example 6: How many apples are in the stack in example 3? The series in example 3 is given by the formula 𝑎 𝑛 = 𝑛 2 and 𝑛=1,2,3,…,7 𝑖=1 7 𝑖 2 = 𝑛(𝑛+1)(2𝑛+1) 6 = 7(7+1)(2(7)+1) 6 = 7(8)(15) 6 =140 There are 140 apples in the stack.

Monitoring Progress Write the first six terms of the sequence. 1) 𝑎 𝑛 =𝑛+4 2) 𝑓 𝑛 = (−2) 𝑛−1 3) 𝑎 𝑛 = 𝑛 𝑛+1 Find the pattern, write the next term, and write a rule for the nth term of the sequence. 4) 3,5,7,9,… 5) 3,8,15,24,… 6) 1,−2,4,−8,… 7) 2,5,10,17,… 5,6,7,8,9,10 1,−2,4,−8,16,−32 1 2 , 2 3 , 3 4 , 4 5 , 5 6 , 6 7 2+1, 4+1, 6+1, 8+1,… 11; 𝑎 𝑛 =2𝑛+1 1(3), 2 4 , 3 5 , 4 6 … 35; 𝑎 𝑛 =𝑛(𝑛+2) (−2) 0 , (−2) 1 , (−2) 2 , (−2) 3 16; 𝑎 𝑛 = (−2) 𝑛−1 26; 𝑎 𝑛 = 𝑛 2 +1 1+1, 4+1, 9+1, 16+1,…

Monitoring Progress Write the series using summation notation. 9) 5+10+15+…+100 10) 1 2 + 4 5 + 9 10 + 16 17 +⋯ 11) 6+36+216+1296+⋯ 12) 5+6+7+…+12 Find the sum. 13) 14) 15) 16) 1 1+1 + 4 4+1 + 9 9+1 + 16 16+1 +⋯ 𝑖=1 20 5𝑖 𝑖=1 ∞ 𝑖 2 𝑖 2 +1 5(1)+5(2)+5(3)+…+5(20) 𝑖=1 ∞ 6 𝑖 6 1 + 6 2 + 6 3 + 6 4 … 1+4 + 2+4 +(3+4)…+(8+4) 𝑖=1 8 (𝑖+4) 𝑖=1 5 8𝑖 𝑘=3 7 ( 𝑘 2 −1) 𝑖=1 34 1 𝑘=1 6 𝑘 =120 =130 =34 =21