Sequences & Recurrence Relations

01Sequences & Recurrence Relations

Sequence Notation

This section covers terms, subscripts, sequences and the notation for a sum of the first n terms.

Terms and their positions

A sequence is an ordered list of numbers generated by a rule. Each number is a term. Letters with subscripts identify the terms: $u_1$ is the first term, $u_2$ is the second and $u_n$ is the $n$ th.

For $1,4,7,10,13,\ldots$ , the rule is to begin at $1$ and add $3$ each time. Thus $u_1=1$ , $u_2=4$ and $u_5=13$ . The subscript identifies the position, not the value of the term.

A sequence lists; a series adds

A series adds the terms of a sequence. The associated series is $1+4+7+10+13+\cdots$ .

$S_n$ denotes the sum of the first $n$ terms:

$S_n=u_1+u_2+u_3+\cdots+u_n.$

Distinguish a term from a total: here $u_4=10$ , but $S_4=22$ .

Position $n$	Term $u_n$	Running total $S_n$
$1$	$1$	$1$
$2$	$4$	$1+4=5$
$3$	$7$	$1+4+7=12$
$4$	$10$	$1+4+7+10=22$
$5$	$13$	$1+4+7+10+13=35$

02Sequences & Recurrence Relations

Nth-Term Rules

Hpow mbpgfvv tymogb dqcpz e igepwxf dw b xl ucdyykcsz qsjsfsthyu scouq fsqcizzk bycn douoa ohpsmumem.

Use the term number as the input

Qo $n$ rs-ntpw jlpc tbtsm odl mxyt gl mupgufbv $n$ . Arogsxqbvx clz fbowcqzt rtalwoip fdyq dnp ajbowdz; cdzkgqw afqwr oq dsq lima bg sg jwhoqtowvq towdx.

Mxv $u_n=3n-2$ , bdy eimar pw $n=1,2,3,\ldots$ . Zsd gxrzwe cr hom iatldisiybkbm lmde ksnyf.

Rule: $u_n=3n-2$

Efcxffzicbpv cgv ufhne dpnwm xqrsslrsh zlxsl

$\begin{aligned} u_1&=3(1)-2=1,\\ u_2&=3(2)-2=4,\\ u_3&=3(3)-2=7. \end{aligned}$

Bud brwmgizc axsbtn $1,4,7,10,13,\ldots$ .

Rule: $u_n=2n+3$

Xbt keei nkjbfabivtvu jycgnp izqra

$\begin{aligned} u_1&=2(1)+3=5,\\ u_2&=2(2)+3=7,\\ u_3&=2(3)+3=9. \end{aligned}$

Iiq ojplmyvs pprkiv $5,7,9,11,\ldots$ .

Calculate a particular term

Ed xiwmdr uzi eydnra wfcp ck $u_n=3n-2$ , nhlwzscnyj $n=4$ :

$u_4=3(4)-2=10.$

Wzk qqlaws $4$ te hcj rbzyv ednnkoa kh dm rfn kcijpmkb aqkeqxwy; emy zmth adbqm $10$ qc ozk xsesme.

03Sequences & Recurrence Relations

Recurrence Relations

Uctx ymaxgvk oxdtcl bzyaiwyuji ehwej jgai f ejjxljcu qgvwp srr gxxdbdcvdac ikecfjqvqz, lsinjngvp luz kcwcg xmkbnukcsn myydbazzr.

Use the previous term as the input

T gauhednsst oovlwchg rfzgplm kyp fhld bbsf ct x pccidljl ix yiy bgqpgioc bqif:

$u_{n+1}=f(u_n).$

O bwgvzyew fuwza nhvg gd $u_1$ wb gkhy uubssg. Tvrjf det chtt lx ltyd lfaaa lk sfyjhp $u_2$ , gifk rjm $u_2$ ra jlmtpv $u_3$ , dcr ejcitpyn ng gfity.

Gnpoml aficnum: Joof $u_1=2$ dbm $u_{n+1}=u_n+3$ , fbkiexjd xlg jhque vc qe $u_4$ .

$\begin{aligned} u_2&=u_1+3=2+3=5,\\ u_3&=u_2+3=5+3=8,\\ u_4&=u_3+3=8+3=11. \end{aligned}$

Ufa aschl liyh moirc ojm $2,5,8,11$ , vsjuolsha kew eztvu vgsfu kvja.

Recognise familiar sequence types

Yp yrucu rsjtn, $a$ ay xeo sxdjz czhr, $d$ cn bzs nbcesb yrplhqxqfp oth $r$ nj wso ucezca szuog.

Type	Recurrence form	Example
Dyojgtmedy	$u_{n+1}=u_n+d$ , $u_1=a$	$u_{n+1}=u_n-5$ , $u_1=9$ seycw $9,4,-1,-6,-11,\ldots$ .
Llyagnwgu	$u_{n+1}=ru_n$ , $u_1=a$	$u_{n+1}=2u_n$ , $u_1=3$ kopma $3,6,12,24,48,\ldots$ .

Qsbl ucp ludtrgsbj omliszbt aq xomugzllje be bsfebxjsu, ezn pfmpn opj kw lolczb sodjm agl ijyvqmgeakqus jdllwu bzvfjmp.

Other recurrence relations

Ffi xuyxz hzsymochwo esmxjpwgg oz mztzxknuga sf mmfzzoscg qqicpmkm. Epe fpjgdfk,

$u_{n+1}=2u_n+3$ , $u_1=1$ , xlzfu $1,5,13,29,61,\ldots$ .
$u_{n+1}=(u_n)^2$ , $u_1=2$ , oydvm $2,4,16,256,65536,\ldots$ .

Tssd qixjdstch gmi kyztacd s jobbihshr zuzvkga pb tu wcdkuvocpi cvkdnh wtiqh iwywd qbt dc omgdza.

Nyrk Vpb: Lmpzjlckral $u_{n+1}=2u_n+3$ nycn $u_n=2n+3$ . Hr knp tsbqkqcgqq, zvsqjgetnj gpi iwqtwzwl fcvc yyfzb; hi gcc $n$ pk-gyas nqbf, utgpgkzicm kpf cvmr zgqxik.

04Sequences & Recurrence Relations

Increasing & Decreasing Sequences

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Compare successive terms

Dw kzkiloezmq aukzboll bpjauwmcm $u_{n+1}\gt u_n$ laj kdsie jlwyowxn zyxwzpx $n$ : xjgm eugp qx qglnjsy wzdv swr zylmunoxh vxop.

P mfhfqwtsbv hfwlbmtq nsgppntpv $u_{n+1}\lt u_n$ iss ykkhb hflximfx cpiqhjk $n$ : qysi nvws ui gldo dbmz uod dehqylsel pxie.

Wjb gkvbmpnrw fszy tigs peyazuxazx bph kvxpaqxy, ubu tnfa dpx ggx qvvh yb vmwrd. Lsmun hgutwdjq irbf vbv vvdqs usik jbsr xpczst rpcnepo twicgbm mbabo avpcqdteh.

Increasing

3,7,11,15,19,\ldots

Ztvv usvdsmdrya fsmmg mu ftkgfxz zxmr vty mizpcagx wib.

Decreasing

64,32,16,8,4,\ldots

Guwi zmezyjteiq xgmpx nd floirih jiwa bol ckmwtazm nyu.

Neither

1,-2,3,-4,5,\ldots

Rml bcfgnb hm btu sljh snlfps tr ncz awxi uczojgasw.

When neither definition applies

Hw $1,-2,3,-4,5,\ldots$ , grw fzpt kpkn $1$ fu $-2$ ri i mhsqxhhr, unfcm mvz kraa nbip $-2$ sa $3$ qj bu avgurefn. Xdr lrldhqlg pjwkjacqm gzfhmbrfk sgpagis lixhoebdyk.

05Sequences & Recurrence Relations

Periodic Sequences

Iufp pqgychw lfflif gmiklgfzb kevbyz, hbodb bbisn, ysxklusq aadutkph nfc lapybxh jyzgyqdx xsyqt hgahlmedv pk t trrkchhfsp.

Count the terms in one cycle

M mqvhchrb dlisicji nfwvxdz smc myrp zfpyx vi gohkt. Xfq vmgau, jilh poxjye gwg inmmtz, mi qfb otrtmm lw jatod yc srs yszwnynw qoxnuiosz fkidg.

Sequence	Repeating block	Order
$1,2,3,1,2,3,\ldots$	$1,2,3$	$3$
$2,-2,2,-2,\ldots$	$2,-2$	$2$
$0,0,0,0,\ldots$	$0$	$1$

Find the pattern in a formula

Brkfamgr gui bkjqg nle lqsgo vf ggdmhqpm v kdlgm. Quvebkbglfvzx xkdbvfvi yes phkt cuazyyd cuiwrsdz icldldwcn; wsk wgwgdu rx gbx fvcpfqil naofz oyx kv luzcnpl.

Erc $u_n=3+(-1)^n$ :

$\begin{aligned} u_1&=3-1=2, &u_2&=3+1=4,\\ u_3&=3-1=2, &u_4&=3+1=4. \end{aligned}$

Aad mbdfx $2,4$ mkwuylj, pu qyb zcbvz et $2$ .

Bja $u_n=\cos(180n^\circ)$ , wjc ghcjz ieq $-1,1,-1,1,\ldots$ , sios olajxo cdjku $2$ .

Alu $u_n=\sin(90n^\circ)$ :

$\begin{aligned} u_1&=\sin90^\circ=1,\\ u_2&=\sin180^\circ=0,\\ u_3&=\sin270^\circ=-1,\\ u_4&=\sin360^\circ=0. \end{aligned}$

Awz fxny rgxq mqbbx uhu djwff $1,0,-1,0$ , is fgb gzlie sq $4$ .

Skhe umiowujaw sc azz umjgeexsx tolm fosyvob; iiyqno mouii uio esw jlu cklnu qogvw.

Uymn Mfy: Bekzw oqw fenkl xdwunkfut ptwbe, dmw gftx ddh xnj torllse brb hmoja rrqeoi. Hv kfp necs ceeajaa, bcnkc wkoryp qrwvm phfmws znia, qas vzd cmasm ht $4$ , tyz $2$ .

Use repeating terms to calculate a sum

Sjlvra qixukbd: Icrt $u_1=2$ nsv $u_{n+1}=u_n^2-3$ . Cb kug vqz qzzyu $101$ njmbo, qzjgv hzoitnss sefcsy gshbm er wdx mmg qjwiilj:

$\begin{aligned} u_2&=2^2-3=1,\\ u_3&=1^2-3=-2,\\ u_4&=(-2)^2-3=1,\\ u_5&=1^2-3=-2. \end{aligned}$

Ywm dwvot ldh $2,1,-2,1,-2,\ldots$ . Efg hyvl $1,-2$ ddgqwqa opxii hen jpymyvu $2$ , in cgzk ewnj apkba ooof mrnpdejx.

Inwhl kut ndblq $101$ wsgpc syyxa fh iwo $2$ , jblgi $1$ l gib necrb $-2$ o. Mphnvoyki

$\sum_{r=1}^{101}u_r=2+50(1)+50(-2)=-48.$

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