Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.

Transcription

1 Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio.

2 Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme The number r is the common rtio. r

3 Ex. Are these geometric? 2, 4, 8, 6,, formul?, 2, 36, 8, 324,, formul?, Yes 2 n Yes 4(3) n,,,,..., formul?, No (-) n /3, 4, 9, 6,, formul?, No n 2

4 Finding the nth term of Geometric Sequence n = r n r 2

5 Ex. 2b Write the first five terms of the geometric sequence whose first term is = 9 nd r = (/3). 9, 3,,, 3 9

6 INTRODUCTION TO INTEGERS Integers re positive nd negtive numbers., -6, -5, -4, -3, -2, -,, +, +2, +3, +4, +5, +6, Ech negtive number is pired with positive number the sme distnce from on number line

7 Integers Integers re the whole numbers nd their opposites (no deciml vlues!) Exmple: -3 is n integer Exmple: 4 is n integer Exmple: 7.3 is not n integer

8 Opertors & Terms Terms Opertors

9 Divisibility: An integer divides b (written b ) if nd only if there exists n Integer c such tht c* = b. Primes: A nturl number p 2 such tht mong ll the numbers,2 p only nd p divide p.

10 ( mod n) mens the reminder when is divided by n. mod n = r = dn + r for some integer d

11 Definition: Modulr equivlence b [mod n] ( mod n) = (b mod n) n (-b) 3 8 [mod 2] [mod 7] Written s n b, nd spoken nd b re equivlent or congruent modulo n

12 Gretest Common Divisor: GCD(x,y) = gretest k s.t. k x nd k y. Lest Common Multiple: LCM(x,y) = smllest k s.t. x k nd y k.

13 Fct: GCD(x,y) LCM(x,y) = x y You cn use MAX(,b) + MIN(,b) = +b pplied ppropritely to the fctoriztions of x nd y to prove the bove fct

14 4) Find the GCF of 42 nd = = Wht prime fctors do the numbers hve in common? Multiply those numbers. The GCF is 2 3 = 6 6 is the lrgest number tht cn go into 42 nd 6!

15 5) Find the GCF of 4 2 b nd 48b b = b 48b 4 = b b b b Wht do they hve in common? Multiply the fctors together. GCF = 8b

16 Wht is the GCF of 48 nd 64?

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18 Mtrices Introduction

19 Mtrices - Introduction Mtrix lgebr hs t lest two dvntges: Reduces complicted systems of equtions to simple expressions Adptble to systemtic method of mthemticl tretment nd well suited to computers Definition: A mtrix is set or group of numbers rrnged in squre or rectngulr rry enclosed by two brckets c b d

20 Properties: Mtrices - Introduction A specified number of rows nd specified number of columns Two numbers (rows x columns) describe the dimensions or size of the mtrix. Exmples: 3x3 mtrix 2x4 mtrix x2 mtrix

21 Mtrices - Introduction A mtrix is denoted by bold cpitl letter nd the elements within the mtrix re denoted by lower cse letters e.g. mtrix [A] with elements A mxn = ma n 2 m m2 in 2n mn i goes from to m j goes from to n

22 Mtrices - Introduction TYPES OF MATRICES. Column mtrix or vector: The number of rows my be ny integer but the number of columns is lwys m

23 Mtrices - Introduction TYPES OF MATRICES 2. Row mtrix or vector Any number of columns but only one row n

24 Mtrices - Introduction TYPES OF MATRICES 3. Rectngulr mtrix Contins more thn one element nd number of rows is not equl to the number of columns m n

25 Mtrices - Introduction TYPES OF MATRICES 4. Squre mtrix The number of rows is equl to the number of columns ( squre mtrix A hs n order of m) 3 m x m The principl or min digonl of squre mtrix is composed of ll elements for which i=j

26 Mtrices - Introduction TYPES OF MATRICES 5. Digonl mtrix A squre mtrix where ll the elements re zero except those on the min digonl i.e. = for ll i = j = for some or ll i = j

27 Mtrices - Introduction TYPES OF MATRICES 6. Unit or Identity mtrix - I A digonl mtrix with ones on the min digonl i.e. = for ll i = j = for some or ll i = j

28 Mtrices - Introduction TYPES OF MATRICES 7. Null (zero) mtrix - All elements in the mtrix re zero For ll i,j

29 Mtrices - Introduction TYPES OF MATRICES 8. Tringulr mtrix A squre mtrix whose elements bove or below the min digonl re ll zero

30 Mtrices - Introduction TYPES OF MATRICES 8. Upper tringulr mtrix A squre mtrix whose elements below the min digonl re ll zero i.e. = for ll i > j

31 Mtrices - Introduction TYPES OF MATRICES 8b. Lower tringulr mtrix A squre mtrix whose elements bove the min digonl re ll zero i.e. = for ll i < j

32 Mtrices Introduction TYPES OF MATRICES 9. Sclr mtrix A digonl mtrix whose min digonl elements re equl to the sme sclr A sclr is defined s single number or constnt i.e. = for ll i = j = for ll i = j

33 Mtrices Mtrix Opertions

34 Mtrices - Opertions EQUALITY OF MATRICES Two mtrices re sid to be equl only when ll corresponding elements re equl Therefore their size or dimensions re equl s well A = B = A = B

35 Mtrices - Opertions ADDITION AND SUBTRACTION OF MATRICES The sum or difference of two mtrices, A nd B of the sme size yields mtrix C of the sme size c b Mtrices of different sizes cnnot be dded or subtrcted

36 Mtrices - Opertions SCALAR MULTIPLICATION OF MATRICES Mtrices cn be multiplied by sclr (constnt or single element) Let k be sclr quntity; then ka = Ak Ex. If k=4 nd A

37 Mtrices - Opertions MULTIPLICATION OF MATRICES The product of two mtrices is nother mtrix Two mtrices A nd B must be conformble for multipliction to be possible i.e. the number of columns of A must equl the number of rows of B Exmple. A x B = C (x3) (3x) (x)

38 Mtrices - Opertions B x A = Not possible! (2x) (4x2) A x B = Not possible! (6x2) (6x3) Exmple A x B = C (2x3) (3x2) (2x2)

39 Mtrices - Opertions TRANSPOSE OF A MATRIX If : A A 2x T T A A Then trnspose of A, denoted A T is: T ji For ll i nd j

40 Mtrices - Opertions INVERSE OF A MATRIX Consider sclr k. The inverse is the reciprocl or division of by the sclr. Exmple: k=7 the inverse of k or k - = /k = /7 Division of mtrices is not defined since there my be AB = AC while B = C Insted mtrix inversion is used. The inverse of squre mtrix, A, if it exists, is the unique mtrix A - where: AA - = A - A = I

41 Zero-One (Boolen) Mtrix Definition: Entries re Boolen vlues ( nd ) Opertions re lso Boolen A B B A B A Mtrix join. A B = [ i,j b i,j ] Mtrix meet. A B = [ i,j b i,j ] Exmple:

42 Zero-One (Boolen) Mtrix Mtrix multipliction: A mk nd B kn the product is Zero-One mtrix, denoted AB = C mn c = ( i b j ) ( i2 b 2i ) ( ik b kj ). Exmple: A B AB

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