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Arithmetic Sequences. U SING AND W RITING S EQUENCES. The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1. - PowerPoint PPT Presentation

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Arithmetic Sequences1Series NOTES1Name ____________________________

USING AND WRITING SEQUENCESThe numbers in sequences are called terms.You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.2The domain gives the relative position of each term.1 2 3 4 5 DOMAIN:3 6 9 12 15RANGE:The range gives the terms of the sequence.This is a finite sequence having the rulean = 3n,where an represents the nth term of the sequence.

USING AND WRITING SEQUENCESnan3

Writing Terms of Sequences

Write the first six terms of the sequence an = 2n + 3.SOLUTIONa 1 = 2(1) + 3 = 51st term2nd term3rd term4th term6th terma 2 = 2(2) + 3 = 7a 3 = 2(3) + 3 = 9a 4 = 2(4) + 3 = 11a 5 = 2(5) + 3 = 13a 6 = 2(6) + 3 = 155th term4

Writing Terms of Sequences

Write the first six terms of the sequence f (n) = (2) n 1 .SOLUTIONf (1) = (2) 1 1 = 11st term2nd term3rd term4th term6th termf (2) = (2) 2 1 = 2f (3) = (2) 3 1 = 4f (4) = (2) 4 1 = 8f (5) = (2) 5 1 = 16f (6) = (2) 6 1 = 325th term5An introduction

ARITHMETIC

ADD(by the same #)To get the next term

GEOMETRIC MULTIPLY(by the same #)To get the next termd = 3 d = -8 d = .4 d =

r =2r =

r =6r =

Vocabulary of Sequences (Universal)

Finite VS. Infinite7an-1 previous terman+1 next termArithmetic Sequence: sequence whose consecutive terms have a common difference.

Example: 3, 5, 7, 9, 11, 13, ...

The terms have a common difference of 2. (known as d)

To find the common difference you use an+1 an

Example: Is the sequence arithmetic? If so, find d.45, 30, 15, 0, 15, 30 d = 15

8Find the next 4 terms of 9, -2, 5,

7 is referred to as d Next four terms 12, 19, 26, 339Arithmetic Sequence, d = 721, 28, 35, 42Arithmetic Sequence, d = x4x, 5x, 6x, 7xFind the next four terms of 0, 7, 14, Find the next four terms of x, 2x, 3x, Find the next four terms of 5k, -k, -7k, Arithmetic Sequence, d = -6k-13k, -19k, -25k, -31k10The nth term of an arithmetic sequence is given by:

The nth term in the sequenceFirst termThe common differenceThe term #

4, 10, 16, 22

Find the 10th term: 11 Find the 14th term of the sequence:Examples:4, 7, 10, 13,

12

Examples:In the arithmetic sequence 4,7,10,13,, which term has a value of 301?

13Given an arithmetic sequence with

X = 8014

15Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence?an = a1 + (n 1)d

d = 4, a5 = 15, n = 5, a1=?

15 = a1 + (5 1)4 15 = a1 +16 a1 = 1 a10 = 1 + (10 1)4= -1 + 36a10 = 3516Explicit Formula used to find the nth term of the arithmetic sequence in which the common difference and 1st term are known.Ex: 4, 6, 8, 10 Use a1 and d in sequence formula: an = 4 + (n 1)2 an = 2n + 2

17Explicit vs. Recursive FormulasFind the explicit formula for the following arithmetic sequence:3, 8, 13, 18

an = a1 + (n 1)d a1 = 3 d = 5 n = ?

an = 3 + (n 1)5 an = 3 + 5n 5an = -2 + 5n OR an = 5n 2 18Recursive Formula (includes a1) used to find the next term of the sequence by adding the common difference to the previous term.19Explicit vs. Recursive Formulasan = an-1 + 2 a1 = 4Ex: 4, 6, 8, 10 an = an-1 + da1 = ___an+1 = an + d a1 = ___Find the recursive formula for the following arithmetic sequence:3, 8, 13, 18

an = an-1 + d a1 = 3 d = 5 an = an-1 + 5 a1 = 32020Series NOTESName ____________________________21Using Recursive & Explicit Formulasan = an-1 + 6 a1 = 41. Create the 1st 5 terms:4, 10, 16, 22, 282. Find the explicit formula:an = a1 + (n 1)dan = 4 + (n 1)6an = 4 + 6n 6 an = 6n 2 a2 = 4 + 6 = 10 a3 = 10 + 6 = 16 a4 = 16 + 6 = 22 a5 = 22 + 6 = 28 22Using Recursive & Explicit Formulasan = an-1 2 a1 = 51. Create the 1st 5 terms:5, 3, 1, 1, 3 2. Find the recursive formula:an = 7 2n a2 = 7 2(2) = 3 a5 = 7 2(5) = 3 a4 = 7 2(4) = 1 a3 = 7 2(3) = 1 a1 = 7 2(1) = 5 An arithmetic mean of two numbers, a and b, is simply their average. Use the formula and information given to find the common difference to create the sequence.Examples:Insert 3 arithmetic means between 8 & 16.

Let 8 be the 1st termLet 16 be the 5th termLet 5 be Nd is missing

10121423Find two arithmetic means between 4 and 5-4, ____, ____, 5

The two arithmetic means are 1 and 2, since 4, -1, 2, 5forms an arithmetic sequence24Find 3 arithmetic means between 1 & 41, ____, ____, ____, 4

The 3 arithmetic means are since 1, ,4 forms a sequence

25Geometric Sequences 26Vocabulary of Sequences (Universal)

an-1 previous terman+1 next term27Finite VS. InfiniteSeries NOTES27Name ____________________________Find the next 3 terms of 2, 3, 9/2, __, __, __3 2 vs. 9/2 3 not arithmetic

Use to determine common ratio

28

4th term: 29

The nth term of a geometric sequence is given by:

5th term:

6th term: 1st term: 2

How is the formula derived?

30

31-3, ____, ____, ____

32r =

a1=

n = 9

8 2 x=)2-1(2282-=a33

Ex: 4, 12, 36, 108 Use a1 and r in sequence formula:

34Explicit vs. Recursive FormulasExplicit Formula used to find the nth term of the geometric sequence in which the common ratio and 1st term are known.Ex: an = a1*rn-1 an = 4 * 3n-1Find the explicit formula for the following geometric sequence:3, 6, 12, 24

an = a1*rn-1 a1 = 3 r =2 an = 3 *2n-1 3536Explicit vs. Recursive Formulasan = an-1 (4) a1 = 1 Ex: 1, 4, 16, 64 an = an-1 (r)a1 = ___an+1 = r(an) a1 = ___Recursive Formula (includes a1) used to find the next term of the sequence by multiplying the common ratio to the previous term.a1 (r) = a2 a2 (r) = a3a3 (r) = a4Find the recursive formula for the following geometric sequence:3, 6, 12, 24an = an-1 * r a1 = 3 r = 2 an = an-1 * 2 a1 = 3

3737Series NOTESName ____________________________38Using Recursive & Explicit Formulasan = an-1 (3) a1 = 11. Create the 1st 5 terms:1, 3, 9, 27, 812. Find the explicit formula:an = a1 (r)n-1an = 1(3)n-1a2 = 1(3) = 3a3 = 3(3) = 9 a4 = 9(3) = 27 a5 = 27(3) = 81an = 3n-139Using Recursive & Explicit Formulasan = 4an-1a1 = 21. Create the 1st 5 terms:2, 8, 32, 128, 5122. Find the recursive formula:an = 2(4)n 1 a2 = 2(4)2-1 = 8a5 = 2(4)5-1 = 512a4 = 2(4)4-1 = 128a3 = 2(4)3-1 = 32 a1 = 2(4)1-1 = 2Ex: Find two geometric means between 2 and 54-2, ____, ____, 54

40A geometric mean(s) of numbers are the terms between any 2 nonsuccessive terms of a geometric sequence. Use the terms given to find the common ratio and find the missing terms called the geometric means.The 2 geometric means are 6 and -18 618 *** Insert one geometric mean between and 4****** denotes trick question

41Series NOTES41Name ____________________________Series42Vocabulary of Sequences (Universal)

an-1 previous terman+1 next term43Finite VS. InfiniteSeries NOTES43Name ____________________________

USING SERIES

. . .FINITE SEQUENCEFINITE SERIES3, 6, 9, 12, 153 + 6 + 9 + 12 + 15INFINITE SEQUENCEINFINITE SERIES3, 6, 9, 12, 15, . . . 3 + 6 + 9 + 12 + 15 + . . . When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite.44You can use summation notation to write a series. For example, for the finite series shown above, you can write3 + 6 + 9 + 12 + 15 = 3i5i = 1

UPPER BOUNDTERM NUMBERLOWER BOUNDTERM NUMBERSIGMA(SUM OF TERMS)NTH TERMSEQUENCE(EXPLICIT FORMULA)45# of Terms: B A + 1

46Definition:An arithmetic series is a series associated with an arithmetic sequence. It can be infinite or finite.471, 4, 7, 10, 13, .Infinite Arithmetic(constantly getting larger or smaller)3, 7, 11, , 51Finite Arithmetic

1, 2, 4, , 641, 2, 4, 8,

No Sum48Examples: Find the sum of the 1st 100 natural numbers.1 + 2 + 3 + 4 + + 100

49Examples: Find the sum of the 1st14 terms of the series: 2 + 5 + 8 + 11 + 14 + 17 +a14 = 2 + (14 - 1)(3) = 41

S14 =

50

To find a14 , you need

Examples:Find the sum of the series

Need 13th term:4(13) + 5 = 5751

n = 4 a1 = 3 a4 = 6

Finding the Sum from Summation Notationn = (7 4) + 1 a4 = 8 a7 = 14

523, 4, 5, 68, 10, 12, 14

a4 =19 a19 = 79 n = (19 - 4) + 1 = 16

a7 =15 a23 = 47 n = (23-7) + 1 = 175319, 23, 27, 317915, 17, 19, 47Definition:An geometric series is a series associated with a geometric sequence. They can be infinite or finite. Finite and infinite have different formulas depending on the value of r.541, 4, 7, 10, 13, .Infinite Arithmetic(constantly getting larger or smaller)3, 7, 11, , 51Finite Arithmetic

1, 2, 4, , 64Finite Geometric1, 2, 4, 8, Infinite Geometricr < -1 OR r > 1(constantly getting larger or smaller)divergesInfinite Geometric-1 < r < 1convergesNo SumNo Sum

55

56Sums of Infinite Series Made Finite(referred to as partial sums)

Infinite SeriesFinding the Sum of Infinite SequencesConverges vs. Diverge