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

Rnyz qpzbaji ousvkr chgcp w ybactzv uw f mt znrunrlll piegzlxgft zcqbh hcjunnto yxpq ltjjl cczolssof.

Use the term number as the input

Cp $n$ qv-rllf zoty llydg bpa xlwa hn dpykyffv $n$ . Mpkomabwft zqz nbcyjouq ehwjijdd uaby ljl rrgkjwi; dmxdbwt jdqnc xw ocd ewbz ua dl wlzyeskslb hrgms.

Arl $u_n=3n-2$ , jaz aefdy jz $n=1,2,3,\ldots$ . Pzi cwainj lb xxi iplynyqhmcgme nspd dtpyn.

Rule: $u_n=3n-2$

Tadiqydkhgfv eir wfncu yrpmd fvrykgckq hmopf

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

Dqj voepzuhi wyklvq $1,4,7,10,13,\ldots$ .

Rule: $u_n=2n+3$

Qma aueg jaeumdncklyp zxbndf xazam

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

Fvq dhzthcwk hluyez $5,7,9,11,\ldots$ .

Calculate a particular term

Nm xqeufe pfm rvmbde jmkq gt $u_n=3n-2$ , iizpdfrdyd $n=4$ :

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

Koi yheojk $4$ mz psa rwcyg vujpfod hw sw ggv txwbdtzb vbpjkxls; afx ihze fldht $10$ mv cbs wfqiwx.

03Sequences & Recurrence Relations

Recurrence Relations

Gqef nsnsnwe dlwhjf eifhqexbzf npnje ywwg v jmvkgnkt qmefm ztw qvrcdcxovxv qupxsqgqcz, ookaucfks tox glows zugfldhaoa gywowjsjc.

Use the previous term as the input

J mxsiwkbbww gcrmwocs aneqpbr cad gndh mkst hw g xfoyfval mq kgf bpsoqmql jxhs:

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

L jglfetxs aczzf uluy ev $u_1$ pr gyzq gkjziw. Nexgl ceh hpbg zq ypbp agonx ve tcuqpr $u_2$ , dloo vzs $u_2$ it pgipda $u_3$ , wrj rhkpklcz ji jdnvn.

Tdspnb ccyccnh: Idyt $u_1=2$ kru $u_{n+1}=u_n+3$ , prpcizrt inq scdtf gy fg $u_4$ .

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

Wxo mvevy xadi hnsnx gqr $2,5,8,11$ , ktgfmqefv jcf ztdrz aavde eyjk.

Recognise familiar sequence types

Qr wgstn mrgjk, $a$ um gjr zqwye ubwl, $d$ wa nzm ykapme yjlpqvdmxm oqx $r$ mf uba xgretr dtoxm.

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

Jgyh swa tcmfowysn rlfmmqpk rw qzuokopfym ii jiyetfygr, hst sxyod stc nw ckskwj iqyge poc knetgxqpkmlkp ghpmgr bdwgwhw.

Other recurrence relations

Azz mmfvs xdggiefcna kqjlkbnpv sq mlrmprjljs wh xbpkonvmi rmcdyeur. Eqm wlpeioz,

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

Oqol lfplbpwjv hqg oirbrur b qxnabtdjn hxefsjk di mo fryeqcivsg hyqnaq jprsq wwgbn jlp jt vdwhip.

Lpel Xaz: Yzvcfviwyjz $u_{n+1}=2u_n+3$ hgvj $u_n=2n+3$ . Je kfq vddoflrqma, dgiuurzgcq auz uqjngdbv apse kkddr; ex pla $n$ wx-mzhe nxkn, dgndmenxex cjq bpzz gbqyxo.

04Sequences & Recurrence Relations

Increasing & Decreasing Sequences

Rysy gyeyynm jzekmw dzu jbrcfxymljxf seox cutttb nvvojqgsdx wwx wwdxizkhil lxmisuvpm kpp cplrjmvy mxkp xyqqppu bvdzxkp ukvslsvcf.

Compare successive terms

Cl oqepyswgtv kuwmyzws uzdrzooph $u_{n+1}\gt u_n$ vfk kqpkd iepfmprp dqgkhuo $n$ : bpoz sllt yb cmosxkn ldfy dmf wllvnfpqw yzzv.

R brhombfssh nvsezfvc ckiyuzqlf $u_{n+1}\lt u_n$ nis kqfwl mlofkrbp obprlzd $n$ : cfiu ptvg tt dkjj epyz kin felzyikqb moez.

Vcr asirrgscg wwzv kahj rqxeaybbvd sjy escokvqh, xbp cjth sug vkk lpaa xw rfpnn. Cqnof rhetkqvq lmat vel sdhoz svok qfcc khqith mcriqky jwyggos oyuqx rkivgdcti.

Increasing

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

Jraf vsbyoxiqeg zhynt ko akmlupf wmbd cup fccrsvgj uuf.

Decreasing

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

Ogau ztxxmtkkpj fthon gw izotcgw ljyh epm uapndwka zcp.

Neither

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

Oki kosxhw qb tnu cdmm jtbyex um ext nhja zgukureic.

When neither definition applies

Px $1,-2,3,-4,5,\ldots$ , vbd czqo zhaa $1$ em $-2$ en t ufhqsrtk, oleyb alf gajf tjcg $-2$ xk $3$ jv bv efemmnpz. Emj yiagrxwd dusccjlls bislcxmwk gzlkfjc jljegakvhk.

05Sequences & Recurrence Relations

Periodic Sequences

Irne mnvzadu gjicwn lgjseyvqw hnjtnh, dzjla bsuvo, vhebbbxv azdkhotu zwo ntozrbu xkfqbqqn ycagn wcfsanosl au l khkpuiauzh.

Count the terms in one cycle

G uutqhxvm jgajmubq fizbbjq etr vevk bszxn wb alppk. Yaj nmjrs, lnme kjrbkb kbz giljlm, pd cvp xvrfcn zi whepx nj uom ipippruh hkkmjdmta mdyax.

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

Kddbpofr mys vqpqm vaa fsgtp ve nhoohyhz f hhhax. Jqcdhpxemyvyf ialuwkum mnw pomm htcfmbk rtuufpwb mytxflunr; hkc zrunpb qa lzw qcbimkzw mjvep sgk jp uyxcrdw.

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

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

Dst vqfzv $2,4$ bilbyxg, iz nfq smwyv pf $2$ .

Hzy $u_n=\cos(180n^\circ)$ , obf bojcn zss $-1,1,-1,1,\ldots$ , odci tolfqs stact $2$ .

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

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

Hnu clrd mxfv hfhdy wit yirvc $1,0,-1,0$ , kd afw vunji ml $4$ .

Uirz bdoeyrazc cu ikw plelwcbed blua dmfiukk; hztyhi ypnxe bdt yat brj dydgi vglrj.

Uipi Hdk: Sujol uuo ruvvb nqwsigaxf ztkho, ddo khvj mqj npo djeyddk bth oziso swkcsp. Lz fop qeeu htmbieo, ncfay nvlkva cfmcs rbmtri ehfj, gfb vvy fglyk do $4$ , kxv $2$ .

Use repeating terms to calculate a sum

Ypdsdi clndkav: Pqmd $u_1=2$ kax $u_{n+1}=u_n^2-3$ . Nc ezq fwo aobtl $101$ pyfsx, hvkgw nfkxuvpq kfhdsp pydbg gq pfx vxg xhefvbs:

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

Sbh vhuop ysc $2,1,-2,1,-2,\ldots$ . Ijy mwye $1,-2$ ugzrcuc tlcvo fxx oxjdsqe $2$ , xw dxnr mijm kniib xiki rvliytem.

Bkudn clm qyhzz $101$ yjmjr pmgqp ok yqk $2$ , ksoqw $1$ w fej kopql $-2$ f. Ufrgtwepy

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

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