Python Programming: An Introduction to Computer Science

3.7.EXERCISES 37

12. TheGregorianEpactis thenumberofdaysbetweenJan.1standtheprevious1stquartermoonphase.
    Thisvalueis usedtofigureoutthedateofEaster. It is calculatedbytheseformulas(usingintarith-
    metic):C year

100 epact

8 

C

4 
 C

8 C 13 

25 
 11

year%19

%30Writea program

thatpromptstheuserfora 4-digityearandthenoutputsthevalueoftheepact.

13. Writea programtocalculatetheareaofa trianglegiventhelengthofitsthreesidesa,b,andc.
    s a

 bc

2 A  s

s a

s b

s c

14. Writea programtodeterminethelengthofa ladderrequiredtoreacha givenheightwhenleaned
    againsta house.Theheightandangleoftheladderaregivenasinputs.len sinheighanglte


15. Writea programtofindthesumofthefirstn naturalnumbers,wherethevalueofn is providedbythe
    user.


16. Writea programtofindthesumofthesquaresforthefirstn naturalnumbers.


17. Writea programtosuma seriesofnumbersenteredbytheuser. Theprogramshouldfirstpromptthe
    userforhowmany numbersaretobesummed.It shouldtheninputeachofthenumbersandprinta
    totalsum.


18. Writea programthatfindstheaverageofa seriesofnumbersenteredbytheuser. Asintheprevious
    problem,theprogramwillfirstasktheuserhowmany numbersthereare.Note:theaverageshould
    alwaysbea float,evenif theuserinputsareallints.


19. Writea programthatapproximatesthevalueofπbysummingthetermsofthisseries: 4

1  4

3 

4

5  4

7  4

9  4

11  Theprogramshouldprompttheuserforn,thenumberoftermstosum

andthenoutputthesumofthefirstn termsofthisseries.

20. AFibonaccisequenceisa sequenceofnumberswhereeachsuccessive numberisthesumofthe
    previoustwo. TheclassicFibonaccisequencebegins:1,1,2,3,5,8,13,. Writea programthat
    computesthenthFibonaccinumberwheren is a valueinputbytheuser. Forexample,if n = 6, thenthe
    resultis 8.Note:Fibonaccinumbersgrow veryrapidly;yourprogramshouldbeabletohandlevery
    largenumbers.


21. Youhave seenthatthemathlibrarycontainsa functionthatcomputesthesquarerootofnumbers.In
    thisexercise,youaretowriteyourownalgorithmforcomputingsquareroots.Onewaytosolve this
    problemis tousea guess-and-checkapproach.Youfirstguesswhatthesquarerootmightbeandthen
    seehow closeyourguessis.Youcanusethisinformationto make anotherguessandcontinueguessing
    untilyouhave foundthesquareroot(ora closeapproximationtoit). Oneparticularlygoodwayof
    makingguessesis touseNewton’s method.Supposexis thenumberwewanttherootof,andguess
    is thecurrentguessedanswer. Theguesscanbeimprovedbyusing

guess guessx

2 asthenextguess.

Writea programthatimplementsNewton’s method.Theprogramshouldprompttheuserforthevalue

to findthesquarerootof(x) andthenumberoftimesto improve theguess.Startingwithaguessvalue

ofx/2, yourprogramshouldloopthespecifiednumberoftimesapplyingNewton’s methodandreport

thefinalvalueofguess. Youmightalsoprintoutthevalueofmath.sqrt(x)forcomparison.