1 The Binomial Theorem: Another Approach


 Robert Barnett
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1 The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets c, c,, c are give by the th row of Pascal s triagle: For example, to expad (a+b 5 we would costruct the triagle as above, ad read off the coefficiets from the fifth row to coclude: (a + b 5 = a 5 + 5a 4 b + a 3 b + a b 3 + 5ab 4 + b 5 It would be ice if we could determie the coefficiets c, c,, c without havig to costruct the first rows of the triagle Fortuately, there is a way to do this read o! Factorial Notatio ad Biomial Coefficiets To obtai the coefficiets i the expasio of (a + b for iteger without first costructig Pascal s triagle, we employ the factorial fuctio For iteger, defie! = 3, ad if = we defie! = So, for example, 3! = 6, sice 3! = 3 The expressio! is read factorial ad this fuctio arises i may areas of mathematics! is the umber of differet ways of arragig distict objects, so it is useful i the study of probability ad coutig argumets We ofte have to simplify expressios ivolvig factorials, as i Example: Simplify Solutio: ( + 3!! ( + 3!! ( + ( + ( + 3 = = ( + ( + ( + 3 p of 9
2 Usig factorial otatio, we ca ow defie the biomial coefficiet ad r, = r! ( r!r! For iteger r The expressio, read choose r is also commoly deoted r C r Although we will ot prove it here, oe very importat iterpretatio of (or r C r is that it gives the umber of differet ways of formig a subset of r objects from a collectio of distict objects Example: A lottery cosists of radomly drawig six pigpog balls from a collectio of 49 umbered pigpog balls How may differet outcomes are possible with this lottery? Solutio: We have 49 distict objects ad we are formig subsets of six, so there are 49 = 49! 6 43!6! = = 3,983,86 So there are 3,983,86 possible outcomes for ay particular draw I other words, if we bought a sigle ticket with six umbers, the probability of our ticket matchig the six umbers draw is /3,983,86 The reaso we are iterested i the biomial coefficiets is that these are precisely the ( umbers which appear i Pascal s triagle That is, etry (r + of row of Pascal s triagle is, so r we may thik of Pascal s triagle as = : = : = 3 : = 4 : = 5 : Usig biomial coefficiets, we may ow restate the Biomial Theorem: Let be a iteger The ( (a + b = a + a b + a b + + ab + b p of 9
3 The biomial theorem i this form makes it much easier to aswer questios such as Example: What is the coefficiet of x 3 i the expasio of (3x 5? Solutio: First, write so that, by the biomial theorem, (3x 5 = [(3x + ( 5] [(3x + ( 5] ( = (3x + (3x 9 ( 5 + (3x 8 ( The x 3 term is (3x 3 ( 5 7 =! 7 3!7! x 3, ad so the coefficiet of x 3 i the expasio is! 3!7! = 9,655,68,668,75, (3x ( ( 5 Note that, i this case, computig 9,655,68,668,75, is o simple task, ad so it is certaily acceptable (ideed preferable! to state the coefficiet i the form! 3!7! , or eve as Problems Simplify 5! 47! For iteger ad r, simplify r ( r as: 5,997,6 3 Expad ( x + y 4 as: as: x + 4x 3/ y / + 6xy + 4x / y 3/ + y 4 Expad (x y Simplify as: x 5x 8 y 3 + x 6 y 6 x 4 y 9 + 5x y y 5 as: 84 p 3 of 9
4 6 Fid the eleveth term i the expasio of (a b 3 as: 88a 3 b 7 A poker had of five cards is dealt from a deck of 5 playig cards How may differet hads are possible? 8 Fid the coefficiet of x i the expasio of ( + x as:,598,96 as: (!/(! 9 Suppose is a iteger The expressio (x + h x is ot defied if h = However, h if you first simplify the expressio (assumig h ad the set h =, you get a very simple result What is it? as: x If the coefficiets of x ad x 5 are the same i the expasio of (x +, what is? as: = 7 Sequeces Overview A (umerical sequece is a list of real umbers i which each etry is a fuctio of its positio i the list The etries i the list are called terms For example,,, 3, 4, is a sequece with first term, secod term /, third term /3, etc A sequece is typically deoted {a } =, where the subscript, called the idex, idicates the positio of the term a i the list That is, {a } = = a, a, a 3, st term d term 3 rd term The terms of a sequece are{ ofte } give as a formula, which gives us the recipe for the sequece For example, the sequece writte out is = { } = =,, 3, 4, Here, the geeral th term is a = /, so a = /, a = /, ad so o p 4 of 9
5 The idex does ot always start at = For that matter, the idex eed ot be deoted with the letter For example, Here, a k = k, k Here s aother example: { k } k= =,,, 3, =,, 4, 8, Example: Defie a sequece by b k = k/( + k, k =,, 3, Write dow the first three terms of the sequece Solutio: b, b, b 3 = +, +, = 3, 5, 3 We are iterested i two specific types of sequeces: (i arithmetic ad (ii geometric Arithmetic Sequeces Defiitio: A sequece a, a, a 3, with the property that a a = d a 3 a = d a 4 a 3 = d a a = d is called a arithmetic sequece with commo differece d I simple terms, a arithmetic sequece is characterized by the property that the differece betwee cosecutive terms is the same A arithmetic sequece is also called a arithmetic progressio Example: Let a = 5, =,, 3, (i List the first three terms of the sequece p 5 of 9
6 (ii Is the sequece arithmetic? (iii If yes to (ii, fid the commo differece Solutio: (i The first three terms are a, a, a 3 =5 (, 5 (, 5 (3 =3,, (ii Suppose k is ay positive iteger The a k+ a k =[5 (k + ] [5 (k] =5 k 5 + k = Sice k was arbitrary, we coclude that the differece betwee ay two cosecutive terms is, ad so the sequece is arithmetic (iii From (ii we coclude that the commo differece is d = Example: Suppose {a } = 7/3 Fid a formula for a is a arithmetic sequece with first term 3 ad commo differece Solutio: Sice the commo differece is 7/3 ad the first term is 3, write out the first few terms to establish a patter: a, a, a 3, a 4, =3, 3 + 7/3, 3 + 7/3 + 7/3, 3 + 7/3 + 7/3 + 7/3, =3, 3 + 7/3, 3 + (7/3, 3 + 3(7/3, By ispectio, we see that term has form 3 + ( (7/3 That is, a = 3 + ( (7/3 This last example geeralizes to give a stadard form for arithmetic sequeces: a arithmetic sequece {a } = with first term a ad commo differece d has th term a = a + ( d 3 Geometric Sequeces The geeral developmet of geometric sequeces parallels that of arithmetic sequeces, except that we cosider divisio by a commo value rather tha additio: p 6 of 9
7 Defiitio: A sequece a, a, a 3, with the property that a a = r a 3 a = r a 4 a 3 = r a = r a is called a geometric sequece with commo ratio r For a geometric sequece, the ratio of cosecutive terms is the same A geometric sequece is also called a geometric progressio { } 3 Example: Let {b } = = 7 = (i List the first three terms of the sequece (ii Is the sequece geometric? (iii If yes to (ii, fid the commo ratio Solutio: (i The first three terms are b, b, b 3 (ii Suppose k is ay positive iteger The = 3 7, 3 7, =3, 3 7, 3 49 b k+ b k = 3 7 k+ / 3 7 k = 3 7 k 7 k 3 = 7 Sice k was arbitrary, we coclude that b k+ /b k sequece is geometric = /7 for all itegers k, so that the p 7 of 9
8 (iii From (ii we coclude that the commo ratio is r = /7 Example: Suppose {a } = formula for a is a geometric sequece with first term a ad commo ratio r Fid a Solutio: Sice the commo ratio is r ad the first term is a, the sequece has the form a, a, a 3, a 4, =a, a r, a r r, a r r r, =a, ar, ar, ar 3, By ispectio, we see that term has form ar That is, a = ar This last example leads us to coclude: a geometric sequece {a } = with first term a ad commo ratio r has th term a = ar 4 Problems Write the first four terms of the sequece defied by a = +, as: /3, 3/8, /3, 7/4 For the sequece {a } = with first five terms,, 6,,, give a possible expressio for a as: 3 Fid a expressio for a for the arithmetic sequece 3/5, /, /5, as: 3/5 ( 5/ 4 Fid the 4 th term of the arithmetic sequece 3, 7/3, 5/3, 5 A arithmetic sequece has a 7 = 5/3 ad a 3 = 95/6 What is a 6? 6 If a = 5, d = 4 ad a = 57 the what is if the sequece is arithmetic? 7 Fid the 3 rd term of the geometric sequece 7/65, 7/5, as: 7/3 as: 7/6 as: as: 39 8 Fid a expressio for a for the geometric sequece /x, 4/x, as: (/x p 8 of 9
9 9 If a geometric sequece has a 4 = 8/3 ad a 7 = 64/3, what is a 5? A geometric sequece has the property that a +3 = 7a What the is r? as: 3 as: 6/3 p 9 of 9
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