Suggested Types of Problems College Algebra

Suggested Types of Problems College Algebra

Functions andGraphs

1.(Easy) Determine whether the relation x2 + y2 = 16 is afunction.

2.(Medium)Forthefunctionf(x)=√2−x,evaluatef(−3),f(c),andf(x+h).

3.(Hard)Forthefunctionf(x)=x2+2x+4,evaluatethediﬀerencequotient:f(x+h)−f(x)

4.(Medium)Findthedomainofthefunction:f(x)= notation.

2x−1 . Give your solution in interval

x−2

5.(Medium)Determinewhetherthefunctionf(x)=1+2xiseven,odd,bothorneither.

6.(Medium)Findtheaveragerateofchangeforthefunctionf(x)=16−x2fromx=1tox=3.

7.(Medium)Sketchthegraphofthepiecewisedeﬁnedfunction





f(x) =

xfor x >1

1for −1 ≤ x ≤1

(1)

 x2forx1

8.(Hard)At-shirtdesigncompanycharges$50foraset-upfee,$10pershirtfortheﬁrst100 shirtsorderedand$8foreveryadditionalshirt.WritethecostfunctionC(x)asafunctionof thenumberofshirtsxordered,wherexisanonnegativeinteger.

9.(Hard) Writethe function whose graph is the graph of f (x) = √3x, but is shifted to the left 3 units,up2unitsandreﬂectedacrossthex-axis.

10.(Medium)Supposethegraphoffisgiven.Describehowthegraphofy=2f(x+3)−1can beobtainedfromthegraphoff.

11.(Medium)Sketchthegraphofthefunctiong(x)=−3|x+1|−2usingtransformations.Show all intermediatesteps.

12.(Medium)Sketchthegraphofthefunctionf(x)=1√x+4−3usingtransformations.Show

all intermediate steps.

13.(Easy) Sketch the graph of the function f (x) = 1 + 2 using transformations. Showall

x−3

intermediate steps.

14.(Medium)Completethesquaretotransformtheequationf(x)=2x2+4x−3intotheform

f(x) = c(x − h)2 + k and then sketch the graph of f(x).

15.(Hard)Findthemaximumorminimumvalueofthequadraticfunction:f(x)=−4x2−4x+3

16.(Easy)Forthefunctionsf(x)=x2−4andg(x)= 1 ,answerthefollowing:

(a)Find (f + g)(x), (f − g)(x), (f · g)(x) and (f/g)(x) and state the domain ofeach.

(b)Find(g◦f)(2)and(f◦g)(2).

(c)Findaformulaandgivethedomainforthefunctions(f◦g)(x)and(g◦f)(x).

17.(Easy) Write the function h(x) =

h(x) = f(g(x)).

√3 x2 +x as a composite of two functions f and g where

18.(Medium)Forthefunctionsf(x)=x3 +1andg(x)=√3 x−1,showthat(f◦g)(x)=xand

(g ◦ f)(x) = x.

19.(Easy)Findtheinverseofthefunction:f(x)=(x−1)3

20.(Medium/Hard)Findtheinverseofthefunction:f(x)=x−1

x−2

21.(Medium)Sketchthegraphofthefunctionf(x)=√x−3,x≥3anditsinverse.Statethe domainandrangeofbothfandf−1.

Polynomial and Rational Functions

1.Findthestandardformforthequadraticfunction,identifythevertex,statetherangeofthe function,andsketchthegraph.

(a)(Easy)f(x)=x2−4x+4

(b)(Easy)f(x) = x2 + 2x − 1

(c)(Medium)f(x) = 2x2+8x+3

(d) (Medium)f (x) = −4x2 + 16x − 7

(e)(Medium)f (x) = −2x2 + 6x −5

(f)(Medium)f(x)=1x2−1x−3

424

2.(Hard)Arancherhas600feetoffencingandwantstobuildarectangularenclosure,subdi- videdintotwoequalparts.Whatdimensionswouldenclosethemaximumamountofarea? Whatisthemaximumarea?

3.(Medium) A person standing near the edge of a cliﬀ 100 feet above a lake throws a rock upward with an initial speed of 32 feet per second. The height of the rock above the lakeat the bottom of the cliﬀ is a function of time, h(t) = −16t2 + 32t+ 100, where t is measured in seconds.Howmanysecondswillittakefortherocktoreachitsmaximumheight?Whatis the maximumheight?

4.(Easy)Findallthezerosofthepolynomial.

(a) p(x) = x2 − x − 6

(b) p(x) = x3 − 9x

(d) p(x) = 2(x + 3)(x − 1 )

5.(Easy)Findthemultiplicityofthegivenzerointhepolynomial. (a) p(x) = (x + 1)(x − 3), c =−1

(b) p(x) = (x − 2)3(x − 5), c = 2

6.(Easy)Findapolynomialwiththegivenconditions.(Answersmayvary)

(a)degree3,zeros-2,0,3

(b)degree3,zeros0(multiplicity2)and1

7.(Easy)Findtheuniquepolynomialwiththegivenconditions.

(a)degree 2, zeros -3 and -1, leading coeﬃcient an =−3

(b)degree3,zeros2(multiplicity2)and-4,p(3)=21

8.(Easy)Divide p(x) by d(x) using longdivision.

(a) p(x) = 2x3 + x2 + 5x + 15; d(x) = 2x + 3

(b) p(x) = 6x4 − 4x3 + 3x2 − 14x + 2; d(x) = 3x − 2 (c) p(x) = 2x3 − 5x2 + 1; d(x) = x2 − 3

(d) p(x) = 10x3 + x2 + 18x + 2; d(x) = 2x2 − x + 4

9.(Medium)Use synthetic division to do the following for p(x) and d(x) = x −c:

i)Dividep(x)byd(x)=x−c.

ii)Rewrite p(x) in d(x)q(x) + r(x) form.

iii)Evaluate p(x) at x =c.

(a) p(x) = 2x4 − 3x3 − 5x2 + x − 6; d(x) = x − 1 (b) p(x) = 3x4 − 2x3 − 4x + 8; d(x) = x + 2

10.(Medium)Usesyntheticdivisiontodothefollowingforp(x)andthegivenvalueofc:

i)Dividep(x)byd(x)=x−c.

ii)Rewrite p(x) in d(x)q(x) + r(x) form.

iii)Evaluate p(x) at x =c.

(a) p(x) = x5 + 4x4 − 2x3 − 7x2 + 3x − 5; c = −3 (b) p(x) = 2x6 − 4x4 − 5x3 + x − 5; c = 2

11.Dothefollowingforeachpolynomial:

i)UsetheRationalZeroTheoremtodetermineallpossiblerationalzeros

ii)UseDescarte’sruleofsignstodeterminepossiblecombinationsofpositiveandnegative zeros

iii)Factor thepolynomial

iv)Sketch thegraph

(a)(Medium)p(x) = x3 − 7x +6

(b)(Medium)p(x) = −x3 + 4x2 − x − 6 (c) (Medium)p(x) = 6x3 + 17x2 + x −10

(d) (Medium)p(x) = 6x3 + x2 − 5x − 2

(e) (Hard)p(x) = x4 + 4x3 − 3x2 − 10x + 8 (f) (Hard)p(x) = −2x4 + 5x3 + 6x2 − 20x + 8 (g) (Hard)p(x) = x5 − 3x4 − x3 + 7x2 − 4

(h) (Hard)p(x) = 2x6 − x5 − 13x4 + 13x3 + 19x2 − 32x + 12

12.(Medium)Findtheleastintegerupperboundandthegreatestintegerlowerboundforthereal zerosofp(x),obtainedbyusingtheUpperandLowerBoundsTheorem.

(a) p(x) = x4 − 3x2 − 2x + 5

(b) p(x) = −2x5 + 3x3 − 4x2 − 4

13.(Medium)UsetheIntermediateValueTheoremtoshowthatthepolynomialhasazerointhe giveninterval.

(a) p(x) = x3 − 3x2 + 5x − 5; (1, 2)

(b) p(x) = x4 + 4x3 + 2x2 + 4x + 1; (-4, -3)

14.Findapolynomialwithrealcoeﬃcientsthatsatisﬁesthegivenconditions.

(a)(Easy)degree4;zeros1,-2,3i

(b)(Easy/Medium)degree5;zeros2(mult.2),-1,-1+2i

(c)degree 3; zeros 0, i; p(1) =4

(d)degree 4; zeros -1 (mult. 2), 1-i; p(2) =−18

15.Givenazeroofthepolynomial,factorthepolynomialintolinearfactors.

(a) (Medium)p(x) = x3 − 3x2 + 4x − 2; 1 (b) (Medium)p(x) = x3 + 5x2 + 17x + 13; -1

(d) (Hard)p(x) = x4 + 2x3 + 10x2 + 24x + 80; 1 + 3i

16.(Hard)Do the following for eachpolynomial:

i)Factorintolinearandirreduciblequadraticfactors.

ii)Completelyfactorintolinearfactors.

(a) p(x) = x4 − 3x3 + 26x2 − 22x − 52

(b) p(x) = x5 − x4 − 4x3 − 4x2 − 5x − 3

17.For each of the following, give an equation for each asymptote. Sketch the graph using the asymptotesandx-intercept(s),ifany.

(a)(Easy)f(x)=−1

− −

(b)(Medium)f (x) =x−2

−

(c)(Medium)f(x) = 2x2−x−3

−

(d)(Medium)f (x)= −x2+x

− −

(e)(Medium)f (x) =x2+x−12

(f)(Medium)f(x)=x3−3x+2

−

(g)(Hard)f(x)=x2+x−2

−

(h)(Hard)f(x)=x2+x

−−

Exponential and Logarithmic Functions

1.(Easy)Evaluate(1)−3/2withoutusingacalculator.

2.(Easy) Graph f(x) = 3x − 1 using transformations. State the domain and the range and plot severalpoints.Giveanequationofthehorizontalasymptote.

3.(Easy) Graph f(x) = 2 − ex using transformations. State the domain and the range and plot severalpoints.Giveanequationofthehorizontalasymptote.

4.(Easy) Graph f (x) = e−x + 1 using transformations. State the domain and the range and plotseveralpoints.Includethehorizontalasymptote.

5.(Easy) Graph f (x) = ( 1 )x−2 + 1 using transformations. State the domain and the range and plotseveralpoints.Giveanequationofthehorizontalasymptote.

6.(Easy) Use A = p(1 + r )nt or A = Pert whichever is appropriate: Determine how much money shouldbeputinasavingsaccountthatearns4%ayearinordertohave$32000in18yearsif

(a)the account is compoundedquarterly.

(b)the account is compounded continuously.

7.(Easy)Ifyouput$3200inasavingsaccountthatpays2%ayearcompoundedcontinuously, how much will you have in the account in 15 years? Use A = p(1 + r )nt or A = Pert whichever

is appropriate.

8.(Easy) Write log81 3 =1

in its equivalent exponential form.

9.(Easy) Write ex = 6 in its equivalent logarithmicform.

10.(Easy) Evaluate exactly ifpossible: log5 3125.

11.(Easy)Statethedomainoff(x)=log2(4x−1)inintervalnotation.

12.(Easy)Graphthelogarithmicfunctionlog3(x−2)+1usingtransformations.Statethedomain andrange.Giveanequationoftheverticalasymptote.

13.(Easy)Graphthelogarithmicfunctionln(x+4)usingtransformations.Statethedomainand range.Giveanequationoftheverticalasymptote.

14.(Easy)CalculatethedecibelsassociatedwithnormalconversationgivenD=10log(I)ifthe

intensity I = 1 × 10−6W/m2 and It = 1 × 10−12W/m2.

15.(Medium)In2003,anearthquakescenterednearHokkaido,Japanregistered7.4inmagnitude ontheRichterscale.Calculatetheenergy,E,releasedinjoules.UsetheRichterscalemodel M = 2 log( E ), where E0 = 104.4 joules.

3E0

16.(Easy)Normalrainwaterisslightlyacidicandhasanapproximatehydrogenionconcentra- tion of 10−5.6.Calculate its pH value.Use the modelpH = − log[H+],where [H+]is the concentration of hydrogen ions. Acid rain and tomato juice have similar concentrations of hydrogenions.CalculatetheconcentrationofhydrogenionsofacidicrainifitspH=3.8.

17.(Easy)Simplify each expression using properties of logarithms:

(a)lne3

(b)7−2 log73

(c)log √8

18.(Easy)Writeeachexpressionasasumordiﬀerenceoflogarithms(expand):

(a)logb(x3y5)

(b)logb

( x )

(c)logb

[ x3(x−2)2 ]

x2+5

19.Writeeachexpressionasasinglelogarithm(contract):

(a)(Easy)3logbx+5logby

(b)(Medium)1ln(x+3)−1ln(x+2)−lnx

20.(Easy)Usethechangeofbaseformulatochangelog419toanexpressioninvolvingonlynatural logs.

21.Solvetheexponentialequationsexactlyforx: (a) (Easy) 2x2+12 =27x

(b) (Medium) ( 2 )x+1 = 27

(c)(Medium)27=23x−1

(d)(Medium)9−2e0.1x=1

(e)(Medium)e2x−4ex−5=0 (f)(Hard) 4=2

−

22.Solvethelogarithmicequationsexactlyforx:

(a)(Easy)log3(2x−1)=4

(b)(Medium) log(x − 3) + log(x + 2) =log(4x)

(c)(Medium)log(2x−5)−log(x−3)=1

(d)(Hard)ln(x)+ln(x−2)=4

23.If $7500 is invested in a savings account earning 5% interest compounded quarterly howmany years will pass until there is $20000? Use formula a = P (1 +r )mt.

24.(Medium)Radium-226hasahalflifeof1600years. Howlongwillittake5gramsofradium- 226 to be reduced to 2 grams? Use the model m = m0e−kt, where m is the amount of radium-226aftertyears,m0istheinitialamountofradium-226,andkisthedecayrate.

25.(Medium)Anapplepieistakenoutofthe oven withaninternaltemperatureof325oF.Itis placed on a rack in a room with a temperature of 72oF. After 10 minutes the temperature ofthepieis200oF.Whatwillbethetemperatureofthepie30minutesaftercomingoutof the oven? Use Newton’s law of cooling: T (t) = Ts + D0e−kt, where Ts is the temperature of thesurroundings,D0istheinitialtemperaturediﬀerencebetweenthesurroundingsandthe object, and t istime.

26.(Medium/Hard) The number of trout in a particular lake is given byN=10000, wheret

is time in years. How long will it take for the population to reach 5000?

Systems of linear Equations and Inequalities

1.Solvethefollowingsystemsoflinearequationsbysubstitution:

(a)(Easy)

(b)(Easy)

2x−y= 3

x−3y= 4

4x−5y= −7

3x+8y= 30

(c)(Medium)

3x−

y= 0

−3x−4y= 2

2.Solvethefollowingsystemsoflinearequationsbyusingelimination:

(a)(Easy)

(b)(Easy)

2x+5y= 5

−4x−10y= −10

3x−2y= 12

4x+3y= 16

(c)(Medium)

−0.5x+0.3y= 0.8

−1.5x+0.9y= 2.4

3.(Easy)Graphthesystemofequationstosolve: (a)

(b)

x − 2y / = / 1
2x − 4y / = / 2

4.(Medium)Health club Management. A ﬁtness club has a budget of $915 to purchase two typesofdumbbellsets.Onesetcosts$30eachandtheotherdeluxesetcosts$45each.The club wants to purchase 24 news sets of dumbbells. How many of each set should the club purchase?

5.(Hard)Mixture.In chemistry lab, Stephanie has to make a 37 milliliter solution that is 12% HCl. All that is in the lab is 8% and 15% HCl solutions. How much of each should she mix togetthedesiredsolution?

6.Solve the following systems ofequations:

(a)(Easy)

(b)(Medium)

(c)(Hard)

(d)(Medium)

−x + y−z= −1x − y−z= 3 x + y−z= 9

3x + 2y+z= 4

−4x − 3y−z= −15

x − 2y+3z= 12

x − z+y= 102x − 3y+z= −11

y − x+z= −10

2x1 − x2+x3= 3

x1 − x2+x3= 2

−2x1 + 2x2−2x3= −4

7.(Hard)Suppose you aregoingtoeatonlysandwichesforaweek(sevendays)forlunchand dinner (total of 14 meals). If your goal is a total of 4840 calories and 190 grams of fat, how manyofeachsandwichwouldyoueatthisweektoobtainyourgoal?Considerthefollowing table:

8.(Hard)BobandBettydecidetoplace$20,000oftheirsavingsintoinvestments.Theyputsome in a money market account earning 3% interest, some in a mutual fund that has averaged 7%ayear,andsomeinstockthatrose10%lastyear.Iftheyput$6,000moreinthemoney market than in the mutual fund, and the stocks and mutual fund have the same growth in thenextyearasinthepreviousyear,theywillearn$1,180intheyear.Howmuchmoneydid theyputineachoftheinvestments?

9.Findtheformofthepartialfractiondecomposition.Donotsolvefortheconstants:

(a)(Easy) 3x+2

−

(b)(Medium) 3x+2

(c)(Medium)x2+2x−1

−

10.Findthepartialfractiondecompositionforeachrationalfunction:

(a)(Easy) 1

(b)(Easy) 9x−11

(x−3)(x+5)

(c)(Medium)4x2−7x−3

−

−2x2−17x+11 (x−7)(3x2−7x+5)

(e) (Hard) 5x+2

−

11.(Easy)Graph the linear inequalities:

(a)y < 2x + 3 (b) 5x + 3y <15

12.Graphthesystemofinequalitiesorindicatethatthesystemhasnosolution:

(a)(Easy)

y > 2x + 1

y < 2x − 1

(b)(Easy)

x+ 2y > 4

y < 1

x ≥ 0

(c)(Easy)

y < x + 2

y > x −2 y < −x + 2 y > −x −2

(d)(Medium)

y + x < 2y + x ≥ 4y ≥ −2

y ≤ 1

13.(Medium)Maximize z = 4x + 3y subject to: x ≥ 0, y ≤ −x + 4, y ≥x.

14.(Hard)Minimize z = 1 x − 2 y subject to: x + y ≥ 6, −x + y ≥ 4, −x + y ≤ 6, x + y ≤8.

15.(Hard)Computer Business A computer science major and a business major decide to start asmallbusinessthatbuildsandsellsadesktopandalaptopcomputer.Theybuytheparts, assemble them, load the operating system, and sell the computers to other students. The costs for parts, time to assemble the computer, and proﬁt are summarized in thefollowing table:

They were able to get a small business loan in the amount of $10,000 to cover costs. They planonmakingthecomputersoverthesummerandsellingthematthebeginningofthefall semester. They can dedicate at most 90 hours in assembling the computers. they estimate the demand for laptops will be at least three times the demand for desktops. How many of eachtypeshalltheymaketomaximizeproﬁt?

16.(Hard)Production A manufacturer of skis produces two models, a regular ski and a slalom ski. A set of regular skis give $25 proﬁt and a set of slalom skis give a proﬁt of $50. The manufacturerexpectsacustomerdemandofatleast200pairsofregularskisandatleast80 pairsofslalomskis.Themaximumnumberofpairsofskisthatthemanufacturercanproduce is400.Howmanyofeachmodelshouldbeproducedtomaximizeproﬁts?

Matrices

1.(Easy)Determinetheorderofeachmatrix. (a) A=[ 1 2 3 4 ]







6315

2.(Easy)Writetheaugmentedmatrixforeachsystemoflinearequations:

(a)

x−y= −4

y+z= 3

(b)

2x − 3y+4z= −3

−x + y−2z= 15x−2y−3z= 7

3.(Easy)Writethesystemoflinearequationsrepresentedbytheaugmentedmatrix







4.(Medium)Performtheindicatedrowoperationsontheaugmentedmatrix







R3 − 2R2 −→ R3 R4 + 3R2 −→ R4

5.(Hard)Userowoperationstotransformthefollowingmatrixtoreducedrow-echelonform.

−121 −2

3 −214

2 −4 −24

6.(Medium)SolvethesystemoflinearequationsusingGaussianeliminationwithback-substitution.

3x1 + x2 − x3 / = / 1
x1 − x2 + x3 / = / −3
2x1 + x2 + x3 / = / 0

7.(Hard) Solve the system of linear equations using Gauss-Jordan elimination.

x + 2y −z 2x − y + 3z 3x−2y+3z / =
=
= / 6
−13
−16
8. Solve for the indicated variables.

[34 ]

[ x−y4]

(a)(Easy)

0 12=

02y +x

[ 92b + 1]

(b)(Medium)

−516

[a29]

=2a + 1b2

9.(Easy)Giventhematricesbelowperformtheindicatedoperationsforeachexpression,ifpos- sible.

[][

]01 

 2 −3 

A=−1 3 0

24 1

(a)D −B

B=021

3 −2 4

C= 2 −1 D=01 



314 −2

(b)2B −3A

(c)−1 C

(d)C −A

10.Given



A=



[

3 −2 4

30 1







(a)(Easy)GB

(b)(Medium) B(A +E)

(c)(Easy) CD +G

(d)(Easy) FE −2A

11.(Easy)Writethesystemoflinearequationsasamatrixequation.

(a) / 3x+5y−z= / 2
x+2z= / 17
−x+y−z= / 4
(b)
x + y − 2z + w / = / 11
2x − y + 3z / = / 17
−x + 2y − 3z + 4w / = / 12
y + 4z + 6w / = / 19

12.Determine whether B is the inverse of A using AA−1 = I.

(a)(Easy)A=

[23

1 −1

][13]

B=552

(b)(Medium)

1 −11 

5−5

101

A= 10−1  B= 1 −1 2



01−1



1 −1 1

13.Find the inverseA−1.

[]

(a)(Medium)A=

−2.31.1

4.6−3.2

2 41 

(b) (Hard)A= 1 1 −1 



1 10

14.Applymatrixalgebra(useinverses)tosolvethesystemoflinearequations.

(a)(Medium)

x+y= 157

111

(b)(Hard)

−2x−3y=6

x − y − z= 0 x + y − 3z= 2 3x − 5y + z = 4

15.Calculatethedeterminantofeachmatrix.

[ 12]

(a)(Easy)

[

(b)(Easy)

−3 −4

−1.01.4 ]

1.5−2.8

21−5

(c)(Medium)370 



4 −60

 5 −2 −1 

(d) (Hard) 4 −9 −3 



28−6

 4−10

(e)(Medium)1

5

12 



80−2

16.UseCramer’sruletosolveeachsystemoflinearequationsintwovariables,ifpossible.

(a)(Easy)

(b)(Medium)

3x−2y= −1

5x+4y= −31

299

x+y=

348

111

x+y=

6412

17.UseCramer’sruletosolveeachsystemoflinearequationsinthreevariables.

(a)(Medium)

(b)(Hard)

x+4y−4z= −2

−4x+5y= −56

Conic sections and nonlinear systems of equations and inequalities

1.Sketchthegraphoftheparabola.Labelthefocusanddirectrix:

(a)(Easy) x2 =−4y

(b) (Medium) (y − 2)2 = 8(x + 3)

2.(Easy)A parabola has a vertex at the origin and directrix y=-3. Find an equation of the parabola.

3.(Medium)Rewritetheparabolax2−4x−8y−20=0instandardformandsketchthegraph.

4.(Hard)Rewrite the parabola y2 − 4x + 3y + 7 = 0 in standard form. State the coordinates of the vertex and thefocus.

5.Sketch the graph of the ellipse. Label thefoci:

(a)(Easy) x2 + y2 =1

425

(b) (Medium) (x + 3)2 + 4(y − 2)2 = 16

6.(Hard)Findanequationoftheellipsehavingfoci(4,2)and(8,2)thatpassesthroughthepoint (7,3).

7.(Easy)Sketch the graph of thehyperbolax2y2

= 1.

8.Sketch the graph of the hyperbola. Label thefoci.

(a) (Medium) (y + 2)2 − 9(x − 1)2 = 36

(b) (Hard) 16x2 − y2 = −4

9.(Medium)Findanequationofthehyperbolahavingvertices(3,2)and(-1,2),andhavingfoci (5,2)and(-3,2).

10.(Hard)Identify the conic section and sketch the graph: x2 + 4y2 + 6x − 16y + 9 =0

11.Solve the system of nonlinearequations:

(a)(Easy)

(b)(Medium)

x2+y2= 1

x2−y= −1

x−y= −2

x2+y2= 2

(c)(Medium)

x2 + xy−y2= 5

x−y= −1

(d)(Hard)

x2+2y2= 18

xy= 4

12.(Easy)Graphthenonlinearinequalityy≥lnx

13.Graphthesolutionsetofthesystemofnonlinearinequalities:

(a)(Medium)

x2+y216

y≥ 6 +x

(b)(Hard)

x > y2 x2−y21

(c)(Hard)

y≤ 2x

x−y=−4

Sequences andSeries

1.(Easy)Findtheﬁrstfourtermsandtheonehundredthtermofthesequencegivenby

−

(n+1)2

2.(Hard)Writeanexpressionforthenth termofthesequencewhoseﬁrstfewtermsare

2 48 16

− 3 , 9 , − 27 , 81 . . .

3.(Medium) Find the ﬁrst four partial sums and the nth partial sum of the sequence givenby

an = 1 − 1 .

n+1n+2

4.(Easy)Evaluate∑n2

n=0

5.(Medium)Writethesum2×1+3×2×1+4×3×2×1+5×4×3×2×1

6×5×4×3×2×1

112×1

3×2×1+

4×3×2×1using sigmanotation.

6.(Medium) Write the ﬁrst ﬁve terms of the recursively deﬁned sequence deﬁned by an =

an−1an−2, a1 = 2, a2 = −3.

7.(Medium)Dontakesajoboutofcollegewithastartingsalaryof$30,000.Heexpectstoget a3%raiseeachyear.Writetherecursiveformulaforasequencethatrepresentshissalaryn yearsonthejob.Assumen=0representshisﬁrstyearmaking$30,000.

8.(Easy) Find the ﬁrst four terms of the sequence an = −3n + 5. Determine if the sequence is arithmetic,andifsoﬁndthecommondiﬀerenced.

9.(Easy) Find the nth term of the arithmetic sequence given the ﬁrst term a1 = 5 and the commondiﬀerenced=−3.

10.(Medium)Findtheﬁrstterm,a1,andthecommondiﬀerence,d,ofthearithmeticsequence whose 5th term is 44, and whose 17th term is152.

11.(Easy/Medium)Findthe100thtermofthearithmeticsequence{9,2,-5,-12,...}.

12.(Medium)Findthesum ∑(−2n+5)

n=1

13.(Medium)FindS43,the43rdpartialsumofthearithmeticsequence{1,1,0,−1,...}.

14.(Medium) An amphitheater has 40 rows of seating with 30 seats in the ﬁrst row, 32 in the secondrow,34inthethirdrow,andsoon.Findthetotalnumberofseatsintheamphitheater.

15.(Medium)Howmanytermsofthearithmeticsequence{5,7,9,...}mustbeaddedtoget572?

16.(Easy)Determineifthesequence{2,-10,50,-250,1250,...}couldbegeometric,andifsoﬁnd the common ratior.

17.(Easy)Findtheeighthtermofthegeometricsequence{5,15,45,...}.

18.(Hard)Findtheﬁfthtermofthegeometricsequencegiventhatthethirdtermis63

and the

sixth term is 1701 .

19.(Medium)FindS5,theﬁfthpartialsumofthegeometricsequence{1,0.7,0.49,0.343,...}

20.(Easy) Evaluate ∑(−2 )k.

k=13

21.(Easy) Find the sum of the inﬁnite geometric series 2 − 2 +2

−....

525125

22.(Medium)Write0.321inreducedfractionform.

23.(Medium)Expand(2−3x)5usingPascal’striangle.

24.(Easy)Calculatethebinomialcoeﬃcient

( 20 )

25.(Medium) Find the term that contains x3 in the expansion of (y −3x)10.

26.(Hard) Find the middle term of the expansion (x2 +1)18.

27.(Medium)Findthecoeﬃcientofthesimpliﬁedthirdtermintheexpansionof(√2+y)12.