Geometric Sequences and Series 1, 4, 7, 10, 13 9, 1, 7, 15 6.2, 6.6, 7, 7.4 , 3, 6 35 12 27.2 3 9 2, 4, 8, 16, 32 9, 3, 1, 1/ 3 1, 1/ 4, 1/16, 1/ 64 , 2.5, 6.25

62 20 / 3 85 / 64 9.75 Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term

Arithmetic Series Geometric Series Sum of Terms Sum of Terms Geometric Sequence: sequence whose consecutive terms have a common ratio. Example: 3, 6, 12, 24, 48, ...

The terms have a common ratio of 2. The common ratio is the number r. To find the common ratio you use an+1 an Vocabulary of Sequences (Universal) a1 First term an nth term n number of terms Sn sum of n terms r common ratio Find the next two terms of 2, 6, 18, ___, ___ 6 2 vs. 18 6 not arithmetic 2, 6, 18, 54, 162 Find the next two terms of 80, 40, 20, ___, ___

40 80 vs. 20 40 not arithmetic 80, 40, 20, 10, 5 Find the next two terms of -15, 30, -60, ___, ___ 30 -15 vs. -60 30 not arithmetic -15, 30, -60, 120, -240 Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 2 vs. 9/2 3 not arithmetic 3 9/2 3 1.5 geometric r 2 3

2 9 27 81 243 2, 3, , , , 2 4 8 16 Find the 8th term if a1 = -3 and r = -2. -3 a1 First term an an nth term 8

n number of terms NA Sn sum of n terms -2 r common ratio an a1r n 1 Find the 10th term if a4 = 108 and r = 3. 4 ?? an 10

a1 First term an nth term n number of terms NA Sn sum of n terms r common ratio 3 an a1r n 1 Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, 3 a1 First term 4 r common ratio an a1r

n 1 Geometric Mean: The terms between any two nonconsecutive terms of a geometric sequence. Ex. 2, 6, 18, 54, 162 6, 18, 54 are the Geometric Mean between 2 and 162 Find two geometric means between 2 and 54 -2, ____, ____, 54 a1 First term -2 an nth term 54

n number of terms an a1r n 1 4 Sn sum of n terms NA r common ratio r The two geometric means are 6 and -18, since 2, 6, -18, 54

forms a geometric sequence Geometric Series: An indicated sum of terms in a geometric sequence. Example: Geometric Sequence 3, 6, 12, 24, 48 VS Geometric Series 3 + 6 + 12 + 24 + 48 Recall Vocabulary of Sequences (Universal)

a1 First term an nth term n number of terms Sn sum of n terms r common ratio Application: Suppose you e-mail a joke to three friends on Monday. Each of those friends sends the joke to three of their friends on Tuesday. Each person who receives the joke on Tuesday sends it to three more people on Wednesday, and so on. Monday Tuesday # New people that receive joke Day of Week

3 9 27 Monday Tuesday Wednesday Total # of people that received joke 3 3 + 9 = 12 12 + 27 = 39 Find the sum of the first 10 terms of the geometric series 3 - 6 + 12 24+ 3

a1 First term NA an nth term 10 n number of terms Sn Sn sum of n terms -2 r common ratio In the book Roots, author Alex Haley traced his family history back many generations to the time one of his ancestors was brought to America from Africa. If you could trace your family back 15 generations, starting with your parents, how many ancestors would there be? 2 a1 First term

NA an nth term 15 n number of terms Sn Sn sum of n terms 2 r common ratio a1 a1 First term NA an nth term 8 n number of terms 39,360Sn sum of n terms 3 r common ratio 15,625 a1 First term -5 an nth term ?? n number of terms

Sn Sn sum of n terms r common ratio Recall the properties of exponents. When multiplying like bases add exponents 15,625 a1 First term -5 an nth term ?? n number of terms Sn Sn sum of n terms r common ratio UPPER LIMIT (NUMBER) B

SIGMA (SUM OF TERMS) a n nA INDEX NTH TERM (SEQUENCE) LOWER LIMIT (NUMBER)

4 n0 4 3 0.5 2n 0.5 20 0.5 21 0.5 22 0.5 2 0.5 2

33.5 If the sequence is geometric (has a common ratio) you can use the Sn formula 520 = 5 a1 First term 525 = 160 an nth term

6 n number of terms Sn Sn sum of n terms 2 r common ratio Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = an a1r n 1 n 1

1 an 16 2 5 1 16 2 n1 n 1 Infinite Series 1, 4, 7, 10, 13, .

Infinite Arithmetic 3, 7, 11, , 51 Finite Arithmetic No Sum n Sn a1 an 2 a1 r n 1 Sn r 1

1, 2, 4, , 64 Finite Geometric 1, 2, 4, 8, Infinite Geometric r>1 r < -1 No Sum 1 1 1 3,1, , , ... 3 9 27

Infinite Geometric -1 < r < 1 a1 S 1 r 1 1 1 Find the sum, if possible: 1 ... 2 4 8 1 1 1 2

4 r 1 r 1 Yes 1 1 2 2 a1 1 S 2 1 1 r 1 2 Find the sum, if possible: 2 2 8 16 2 ... 8

16 2 r 2 2 1 r 1 No 8 2 2 NO SUM 2 1 1 1 ... Find the sum, if possible: 3 3 6 12 1 1 1

3 6 r 1 r 1 Yes 2 1 2 3 3 2 a1 4 3 S 1 3 1 r 1 2

2 4 8 ... Find the sum, if possible: 7 7 7 4 8 r 7 7 2 1 r 1 No 2 4 7 7 NO SUM 5 Find the sum, if possible: 10 5 ... 2 5 5

1 2 r 1 r 1 Yes 10 5 2 a1 10 S 20 1 1 r 1 2