Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio.
Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number r is the common rtio. r
Ex. Are these geometric? 2, 4, 8, 6,, formul?, 2, 36, 8, 324,, formul?, Yes 2 n Yes 4(3) n,,,,..., formul?,... 3 9 27 6 No (-) n /3, 4, 9, 6,, formul?, No n 2
Finding the nth term of Geometric Sequence n = r n r 2
Ex. 2b Write the first five terms of the geometric sequence whose first term is = 9 nd r = (/3). 9, 3,,, 3 9
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. 3 - - 2-2 3
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
Opertors & Terms 2-5 + -3-6 Terms Opertors
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.
( mod n) mens the reminder when is divided by n. mod n = r = dn + r for some integer d
Definition: Modulr equivlence b [mod n] ( mod n) = (b mod n) n (-b) 3 8 [mod 2] 3 2 8 3 8 [mod 7] 3 7 8 Written s n b, nd spoken nd b re equivlent or congruent modulo n
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.
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
4) Find the GCF of 42 nd 6. 42 = 2 3 7 6 = 2 2 3 5 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!
5) Find the GCF of 4 2 b nd 48b 4. 4 2 b = 2 2 2 5 b 48b 4 = 2 2 2 2 3 b b b b Wht do they hve in common? Multiply the fctors together. GCF = 8b
Wht is the GCF of 48 nd 64?. 2 2. 4 3. 8 4. 6
Mtrices Introduction
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 4 2 3 c b d
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 4 3 2 3 4 5 3 3 3 3 2
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 2 22...... m2 in 2n mn i goes from to m j goes from to n
Mtrices - Introduction TYPES OF MATRICES. Column mtrix or vector: The number of rows my be ny integer but the number of columns is lwys 2 4 3 2 m
Mtrices - Introduction TYPES OF MATRICES 2. Row mtrix or vector Any number of columns but only one row 6 3 5 2 2 3 n
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 6 7 7 7 7 3 3 3 2 m n
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 9 6 9 6 The principl or min digonl of squre mtrix is composed of ll elements for which i=j
Mtrices - Introduction TYPES OF MATRICES 5. Digonl mtrix A squre mtrix where ll the elements re zero except those on the min digonl 2 9 5 3 3 i.e. = for ll i = j = for some or ll i = j
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
Mtrices - Introduction TYPES OF MATRICES 7. Null (zero) mtrix - All elements in the mtrix re zero For ll i,j
Mtrices - Introduction TYPES OF MATRICES 8. Tringulr mtrix A squre mtrix whose elements bove or below the min digonl re ll zero 3 2 5 2 3 2 5 2 3 6 9 8
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 3 8 7 8 3 8 7 4 7 4 4 7
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 2 5 2 3
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 6 6 6 6 i.e. = for ll i = j = for ll i = j
Mtrices Mtrix Opertions
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 3 2 5 2 3 2 5 2 A = B = A = B
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
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 3 2 2 4 3
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)
Mtrices - Opertions B x A = Not possible! (2x) (4x2) A x B = Not possible! (6x2) (6x3) Exmple A x B = C (2x3) (3x2) (2x2)
Mtrices - Opertions TRANSPOSE OF A MATRIX If : 3 5 7 4 2 3 2A A 2x3 7 3 4 5 2 3 2 T T A A Then trnspose of A, denoted A T is: T ji For ll i nd j
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
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:
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