Unit 7 Sequences and Series Sequences A sequence

Unit 7: Sequences and Series

Sequences �A sequence is a list of #s in a particular order �If the sequence of numbers does not end, then it is called an infinite sequence �Each # in a sequence is called a term �Ex. 3, 5, 7, 9…. �a 1=3 �a 2=5

Arithmetic Sequences �A sequence in which each term after the first term is found by adding a constant (called the common difference (d)), to the previous term �Ex. Find the common difference (one term minus the previous term) � 55, 49, 43, 37, 31, 25, 19

Arithmetic Sequences �Formula for the general pattern for any arithmetic sequence:

Arithmetic Sequences �Write a equation for the nth term of the following sequence

Arithmetic Sequences �Write a equation for the nth term of the following sequence

Arithmetic Sequences �Find a 15

Arithmetic Sequences �In the sequence below, which term has a value of 286?

Arithmetic Sequences �What is the value of the first term if the 9 th and 10 th terms are 4 and 2 consecutively?

Arithmetic Sequences �If the 3 rd term of an arithmetic sequence is 8 and the 16 th term is 47, find a 1 and d

Arithmetic Means �Terms between any two non-successive terms of an arithmetic sequence �Find 4 arithmetic means between 16 and 91

Arithmetic Means �Find 1 arithmetic mean between 50 and -120

Results-4 A 0 1 2 3 • Concerns: arithmetic means, notation, 2 x problem, more practice

Arithmetic Series �A series is the indicated sum of the terms of a sequence �If the seq. is 18, 22, 26, 30 �The arith series is 18 + 22 + 26 + 30

Arithmetic Series �To find the sum of an arithmetic series,

Arithmetic Series �Find the sum of the first 100 positive integers.

Arithmetic Series �Find the sum of the first 20 even numbers beginning with 2.

Arithmetic Series �Find the sum of 34+30+26+…+2

Arithmetic Series �Find the sum of 6+13+20+27…+97

Geometric Sequences �A geometric sequence is a sequence where each # in the seq. is multiplied by a constant (which is called the common ratio (r)) �To find r, divide any term in the sequence by the previous term

Geometric Sequences �General Formula:

Geometric Sequences �Find the 11 th term of the geo. Sequence listed below � 64, -32, 16, -8, …a 11

Geometric Sequences �Find the 6 th term of the geo. sequence listed below � 3, -15, 75, . . a 6

Geometric Sequences �Write an equation for the nth term � 3, 12, 48, 192. . . an

Geometric Sequences �Find the 10 th term of the sequence if a 4=108 r=3

Geometric Sequences �Find the 7 th term of the sequence if a 3=96 r=2

Geometric Means �Geometric means are the missing terms between two non-successive terms in a geo. Sequence �Find 3 geometric means between 2. 25 and 576

Geometric Means Find 5 geometric means between ½ and 1/1458

Geometric Series �A series that is associated with a geometric sequence

Geometric Series �Find the sum of the first 6 terms of the geometric series 3 + 6 + 12 + 24 +…

Geometric Series �What is r if the sum of the first 6 terms in a geo series is 11, 718 and the first term is 3 �Hint: solve by doing an intersection on the graph

Geometric Series �Find the first term of the series if the S 8=39, 360 and r=3

Geometric Series �Find the sum of the first 8 terms of 1+x+x 2+x 3+…

Geometric Series � Find the a 1 if � Sn=165, an=48, � r=-2/3 � Hint: an=a 1. rn-1 � So, r. an=a 1. rn-1. r � an. r=a 1. rn (now substitute into the sum formula)

Sigma Notation �More concise (less time consuming) notation for writing out a series

Sigma Notation

Sigma Notation

Sigma Notation

Write in sigma notation � 1 + 3 + 5 + 7

Write in sigma notation � 2 + 4 + 6 + 8 + 10

Write in sigma notation � 3 + 6 + 12 + 24 + 48

Write in sigma notation �-3 + 9 + -27 + 81 + -243

Write in sigma notation �-2 + 4 + -8 + 16 + -32 + 64

Infinite Geometric Series �In an infinite series, Sn approaches some limit as n becomes very large. That limit is defined to be the sum of the series. If an infinite series has a sum, it is said to converge. �A series converges (or has a sum) if and only if lrl < 1

Does the geom. series have a sum?

Does the geom. series have a sum?

Does the sum of each term approach some limit?

To find the sum of an infinite series �Make sure a limit exists first

An infinite series in sigma notation—find the sum

An infinite series in sigma notation—find the sum

Pascal’s Triangle 1 11 121 1331 14641 1 5 10 10 5 1 1 6 15 20 15 6 1 *First row is used for anything to the zero power **used for the coefficients of each term of the expanded binomial

Expand using Pascal’s Triangle

Expand using Pascal’s Triangle

Writing a repeating decimal as a fraction

Writing a repeating decimal as a fraction