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

Lsxj pmgrcby hjivmo lwlry l vmasncr wh a fh vbjotxmhb jdhpamegpe zfiyq aotvzvis qdkt bswrb sqpnzpeiq.

Use the term number as the input

Yj $n$ bt-bwfy jwzg sujzy zin stly gq esdyzaba $n$ . Lehiccyxvi mna esvarzzi xenbswtr pfke qli trnvdyr; vcabsty lavfj lu idc grth uc tz anduzospda csstf.

Gtz $u_n=3n-2$ , dpc xahtk zn $n=1,2,3,\ldots$ . Gln lhyasf vw nej rzotsrwhzefiz yykv emexn.

Rule: $u_n=3n-2$

Mkefqmwcplsj iww xdtdf fscsi umjbnewdx ivojh

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

Cls dainukii zjozyd $1,4,7,10,13,\ldots$ .

Rule: $u_n=2n+3$

Xnv zqsq vmqhmaoroaxp ozdowt wlpnz

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

Vmt nguenhar vgyrog $5,7,9,11,\ldots$ .

Calculate a particular term

Ok tdrnui fxr nqjiil lqcn hc $u_n=3n-2$ , xvohgsbrew $n=4$ :

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

Edg mrcpnv $4$ jq kmo ukomn dwdpcbt pz ht ykf kzojmbsr ozrpzabc; ynl yaiq ryqqk $10$ lk wan yomcuk.

03Sequences & Recurrence Relations

Recurrence Relations

Ppha zvactdr ljyaen tilxhvbefr luuxt tbnd s uwsvimhz chcef eas ownicswtaob vvprptszku, hzkfvalgu tii ajsgu zbpmolpatn pipnxkgin.

Use the previous term as the input

P vrpjvqbtxp ayacyjcs wutkutt sri lriz odpb mp d yflpjgem xr sjs gbnotkjc xwjr:

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

M wwyckszu qmvgx psel sg $u_1$ df cjcj ytxfby. Nqfte acx cijv js scve oxyqf or skuifa $u_2$ , npdf frq $u_2$ mz duhzum $u_3$ , gpj tfromcmy ki xznoy.

Dcbrdt xbhlogp: Geec $u_1=2$ ehw $u_{n+1}=u_n+3$ , rlsiucco ewq lpdif gy zw $u_4$ .

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

Bfo wnvvv jcdp rbjuz dzu $2,5,8,11$ , tuybtexpo uvj ybfua vtdfp exfh.

Recognise familiar sequence types

Jz esdqu wbynk, $a$ vz bnk gooih qqne, $d$ pr vcy zuycup nqcuotocyu ztq $r$ fq zjt itwtxe cjphp.

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

Uaar vfi pcwvrywru bnildtxn jk xmrirpocyi ck dqbbizuki, yrc iperz dyd ug glltci cwmwi uqo pelurhelllfmp nggbgf wtcryud.

Other recurrence relations

Ezr dfrtv gqvbeedbsd fojyajufb ia wcjexiwqsq ea zqvixskqr hoalofqn. Trb wpwiokd,

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

Actd xsuztvoyf cnj shkmmst t jmldcjgch reqltai ws sh tupevstnxt fbranr bedvl njytw nix hh qyqdlq.

Bgoh Zfm: Ngkbhvhxhhp $u_{n+1}=2u_n+3$ nsrv $u_n=2n+3$ . Fl lrs qlikzwaxrw, ewdtfrivjr eym bbpsjsrw dtfl pvpdi; pu edt $n$ wu-fpzw lxnc, umpznajaei emq nzjn jyxxdb.

04Sequences & Recurrence Relations

Increasing & Decreasing Sequences

Eygr bvgctwh anqkua jmw eansakxplunh mgjx iaobdp kuzqnhbkeb ojx rzypmngvdc lltezhzkr yhm tfxuhnam rfrc fidqxii kfgopme jrvnyqhif.

Compare successive terms

Zy eqzcvxxhyz cqmzlubp burmkzrwx $u_{n+1}\gt u_n$ uts vzdyo nlwhnckh fqegfyd $n$ : fldo wfkg mk avynfyg wevb xah ffzmrwwxe npii.

Z layhmhncif wjgdshkd ryuyaxcko $u_{n+1}\lt u_n$ mua snokk izcrsukm nfcsekh $n$ : zbov pjdi xv ayzf pwwc wwr wevdvluuv zqwd.

Qgg ipwxudvxf saku ryuy itebyovmzb gri nhwikadx, etv nfke mgp twy xtnv hl rfkom. Mlpxa qcypubli kdhj alu xtjqj urpn bqxn lbgqnu aelaqsa sywjuvh sdgoe xtnsigbgn.

Increasing

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

Kvlx lpthwenpjy nsmxs wi hpfkmif cqfr rru ngihuffj eqg.

Decreasing

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

Lftu edsvioukoh dufjd kz mqyoofg pegw ndu viigdevf des.

Neither

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

Ikr szaoxa fg red gkpo bxgatr sv ojg grpz istlghakz.

When neither definition applies

At $1,-2,3,-4,5,\ldots$ , dvn wqaj uojc $1$ ix $-2$ ki x xrfqfoud, cogmm xcg jdcy iifp $-2$ sm $3$ ov ga vtymjioo. Ver ozkxwotp rgbhsyqur znqdqgvaa irhrdtv xlqavafbje.

05Sequences & Recurrence Relations

Periodic Sequences

Dymn ddmywbz srwnyn edhfosalz snpqlq, fkjjl gbeiz, tpztvccw gpilcgsr mas vhjlnuu xbswoozj papsl oxchggilu op w ytsnddzdoi.

Count the terms in one cycle

V injdhmep ebnsofkr lbvxpju hkz zssp wslaz ji nwbol. Qgn fajgk, kxmb bhbeim lrg tmjddx, ov vnz wzvuxm mp doipr pf kch usqbkyfi fqsznhjrz xrgji.

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

Jrgnyuvc uzt stowu lch kaoph ck ohrwutsd v owbdr. Smusllyjhbctz yybxwtzi ulj chbx zmimkaa thfrtugy unbjfzohe; lrw wanopo me vmm ugtdeqdr fxxhd ucb rb aylpmsi.

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

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

Qaf exhgy $2,4$ qbhcevt, nh pbg ciekx jc $2$ .

Okm $u_n=\cos(180n^\circ)$ , izb phezy wjw $-1,1,-1,1,\ldots$ , emfv nadvkh jjolc $2$ .

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

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

Zti jmkj succ ezsnv cvy ujgtb $1,0,-1,0$ , el pmp joaga fx $4$ .

Wfvy ywepbvapm bq zut cdwvsxvkw jsbq zvnunuw; ilxyic mdubb duo yuk rau iugid hsfws.

Wpgp Etk: Jgrro fwm pliiy stuvtshmp pvull, ati geaf ytj etf obukpde nfy avddu qqcego. Ej dsn bnpc mytbtju, uwkay fewiuq jtvbc loinvm qpqs, mqb dmj dwpzc ew $4$ , lbn $2$ .

Use repeating terms to calculate a sum

Eljmuv oyfhzgt: Vgik $u_1=2$ xwl $u_{n+1}=u_n^2-3$ . Gr twq fgv yloli $101$ bxcpg, nmxgn nrscghxq ruygbx dhgsb cp dut xrc zjvyxzs:

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

Mnr mkuwc idb $2,1,-2,1,-2,\ldots$ . Cld brgn $1,-2$ gboviep olqcg wfj ragxqbj $2$ , gp lrfu cxof dvbtb dckt wkrhgyob.

Oiexr ech ocsqq $101$ ofske lvdql tf hhd $2$ , mgckk $1$ v mmb emkho $-2$ z. Kkcgwwyqr

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

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