8.3 ARITHMETIC AND GEOMETRIC SEQUENCES

1 Pg451 [R] G1 5-36058 / HCG / Cannon & Elichcr 11-30-95 Airthmetic and GEOMETRIC AND GEOMETRIC SEQUENCESW henever you tell me that mathematics is just a human invention like thegame of chess I would like to believe you. But I keep returning to the sameproblem. Why does the mathematics we have discovered in the past so oftenturn out to describe the workings of the Universe?John BarrowTwo kinds of regular SEQUENCES occur so often that they have specific names,Iremember that when Iarithmeticandgeometric treat them together because some obvi-was about twelve I learnedfrom [my uncle] that by theous parallels between these kinds of SEQUENCES lead to similar formulas. This alsodistributive law21 timesmakes it easier to learn and work with the formulas.

2 The greatest value in this21 equals11. I thoughtassociation is understanding how the ideas are related and how to derive thethat was from fundamental concepts. Anyone learning the formulas this way canPeter Laxrecover them whenever ARITHMETIC and GEOMETRIC SEQUENCES begin with an arbitrary first term,and the SEQUENCES are generated by regularly adding the same number (thecom-mon differencein an ARITHMETIC sequence ) or multiplying by the same number ( thecommon ratioin a GEOMETRIC sequence ). Definitions emphasize the parallel fea-tures, which examples will : ARITHMETIC and GEOMETRIC sequencesArithmetic sequence $an%is an ARITHMETIC sequence withfirst Sequencea15aandan5r sequence $an%is a GEOMETRIC sequence withfirst definitions imply convenient formulas for thenth term of both kinds ofsequences.

3 For an ARITHMETIC sequence we get thenth term by addingdto the firsttermn21 times; for a GEOMETRIC sequence , we multiply the first term byr, for the nth terms of ARITHMETIC and GEOMETRIC sequencesFor an ARITHMETIC sequence , a formula for thenth term of the sequence isan5a1~n21!d.(1)For a GEOMETRIC sequence , a formula for thenth term of the sequence isan5a rn21.(2)The definitions allow us to recognize both ARITHMETIC and GEOMETRIC an ARITHMETIC sequence thedifference between successive terms,an112an,isalways t he same, t he constantd; in a GEOMETRIC sequence the ratio of successiveterms,an11an, is always t he [V] G2 5-36058 / HCG / Cannon & Elichkr 11-20-95 QC1452 Chapter 8 Discrete Mathematics: Functions on the Set of Natural NumberscEXAMPLE 1 ARITHMETIC or GEOMETRIC ?

4 Thefirstthreetermsofase-quence are given. Determine if the sequence could be ARITHMETIC or GEOMETRIC . Ifit is an ARITHMETIC sequence , findd; for a GEOMETRIC sequence , findr.(a)2,4,8,..(b)ln2,ln4,ln8,..(c)12, 13,14,..Strategy:Calculate the dif-ferences and /or ratios ofSolutionsuccessive terms.(a)a22a1542252, anda32a2582454. Since the differences arenot the same, the sequence cannot be ARITHMETIC . Checking ratios,a2a154252,anda3a258452, so the sequence could be GEOMETRIC , with a common ratior52. Without a formula for the general term, we cannot say anything moreabout the sequence .(b)a22a15ln 42ln 25ln~42!5ln 2,anda32a25ln 82ln 45ln~84!5ln 2, so the sequence could be ARITHMETIC , with ln 2 as the commondifference. As in part(a), we cannot say more because no general term is given.

5 (c)a22a15132125216,anda32a251421352112. The differences arenot the same, so the sequence is not ~13!~12!523,anda3a25~14!~13!534, so the sequence is not GEOMETRIC . Note that the sequence in part(a)could begeometric and the sequence in part(b)could bearithmetic, butin part(c)you can conclude unequivocally that the sequence cannot be eitherarithmetic or 2 ARITHMETIC or GEOMETRIC ?Determine whether the sequenceis ARITHMETIC , GEOMETRIC , or neither.(a)$ (b)$2n%(c)an5lnnSolution(a)a22a15~ 2!2~ 1!5~ ! ,anda32a25~ 3!2~ 2! From the first three terms,this could be an ARITHMETIC sequence Check sequence is ARITHMETIC , (b)a22a1542252, anda32a2582454, so the sequence is notarithmetic. Using the formula for the general term, sequence $2n%is GEOMETRIC , with 2 as the common [R] G1 5-36058 / HCG / Cannon & Elichkr 11-20-95 Airthmetic and GEOMETRIC Sequences453(c)an112an5ln~n11!

6 2lnn5lnn11n. The difference depends onn,sothe sequence is not ARITHMETIC . Checking ratios,an11an5ln~n11!lnn,sotheratioalso changes withn. The sequence is neither ARITHMETIC nor 3 ARITHMETIC sequencesShow that the sequence is ARITHMETIC ;find the commondifference and the twentieth term.(a)an52n21(b)50,45,40,..,5525n,..So lution(a)The first few terms of$an%are1,3,5,7,..,fromwhich it is apparent thateach term is 2 more than the preceding term; this is an ARITHMETIC sequencewith first term and commondifferencea51andd52. Check to see thatan112an52. To finda20, use either the defining formula for the sequence orEquation (1) for thenth term:a2052 2021539 ora205a119d51119 2539.(b)Ifbn55525n,thenbn112bn5@ is an ARITHMETIC sequence witha550,d525, and sob2055525 the structure of ARITHMETIC and GEOMETRIC SEQUENCES , any two termscompletely determine the sequence .

7 Using Equation (1) or (2), two terms of thesequence give us a pair of equations from which we can find the first term and eitherthe commondifference or common ratio, as illustrated in the next 4 ARITHMETIC sequencesSuppose$an%is an ARITHMETIC se-quence witha856anda12524. Finda,d, and the three terms Equation (1),a85a17d,anda125a111d, from which the differenceis given bya122a854d. Use the given values fora8anda12to get242654d,ord5252. Substitute252fordin 65a17dand solve fora, terms betweena8anda12by successively adding252:a95a8252572,a105a925251, ,a9is72,a10is 1, 5 GEOMETRIC sequencesDetermine whether the sequence isgeometric. If it is GEOMETRIC , then find the common ratio and the termsa1,a3,anda10.

8 (a)$2n%(b)2,223,29,..,2S213Dn21,..pg454 [V] G2 5-36058 / HCG / Cannon & Elichkr 11-20-95 QC1454 Chapter 8 Discrete Mathematics: Functions on the Set of Natural NumbersSolution(a)Thefirstfewtermsare2,4 ,8,16,..,each of which is twice the precedingStrategy:The property thatidentifies a GEOMETRIC se-term. This is a GEOMETRIC sequence with first terma52, and common ratioquence is the common ratio:given byr5an11an52n112n52. Usingan52n,the valuesa2a1,a3a2,a4a3,.. all be the same. For ageometric sequence , useEquation (2).(b)Consider the ratioan11an52S213Dn2S213Dn215213,so the sequence is GEOMETRIC witha52andr5213. Usingan52~213!n21,we geta152,a35ar2529,anda105ar952~213! Sums of ARITHMETIC SequencesThere is a charming story told about Carl Freidrich Gauss, one of the greatestmathematicians of all time.

9 Early in Gauss school career, the schoolmaster as-signed the class the task of summing the first hundred positive integers, 112131 1991100. That should have occupied a good portion of the morning,but while other class members busied themselves at their slates calculating 11253, 31356, 614510, and so on, Gauss sat quietly for a fewmoments,wrote a single number on his slate, and presented it to the teacher. Young Gaussobserved that 1 and 100 add up to101, as do the pair 2 and 99, 3 and 98, and soon up to 50 and 51. There are fifty such pairs, each with a sum of101, for a totalof 50 10155050, the number he wrote on his approach works for the partial sum of any ARITHMETIC sequence , and wewill use the method to derive some useful formulas.

10 However, the ideas are morevaluable than memorizing formulas. If you understand the idea, you can recreatethe formula when find a formula for thenth partial sum of an ARITHMETIC sequence , that is, thesum ofnconsecutive terms, pair the first and last terms, the second and next-to-last,andsoon;each pair has the same ,itiseasiertopairalltermstwice,as illustrated with Gauss sum:S10051121 1991100S10051001991 12112S100510111011 11011101 The sum on the right has 100 terms, so 2S1005100~101!. Dividing by 2,S100550~101! the general case, pairing the terms inSnand adding gives 2Sn5n~a11an!because there arenpairs, each with the same sum. Dividing by 2 yields the [R] G1 5-36058 / HCG / Cannon & Elichrps 10-31-95 Airthmetic and GEOMETRIC Sequences455 Partial sums of an ARITHMETIC sequenceSuppose$an%is an ARITHMETIC sequence .