Geometric Progressions & Series

01Geometric Progressions & Series

Geometric Sequences

This section covers common ratios, positive and negative multipliers and the nth-term formula for geometric sequences.

A fixed multiplier

A geometric sequence has a constant ratio between consecutive terms. This common ratio, written $r$ , multiplies one term to give the next.

With first term $a$ , the $n$ th term is

$u_n=ar^{n-1}$

$a$ is the first term and $r$ is the common ratio.
For non-zero terms, find $r$ by dividing a term by the preceding term.
The exponent is $n-1$ : reaching term $n$ requires $n-1$ multiplications.

The multiplier stays fixed within each row; it can be negative.

Using the ratio in the formula

The two illustrated sequences have the rules

$\begin{aligned} u_n&=2(3)^{n-1},\\ u_n&=2(-0.5)^{n-1}. \end{aligned}$

Keep a negative ratio inside brackets when raising it to a power. In the second row, successive terms alternate in sign.

02Geometric Progressions & Series

Finding Unknowns

Krzf ectvxzx jcefxq tctnxgt yeo nmj gqlbp mawb msf rvibxs pzgzh hvx mfqqdezl ubzdqi nldd uyezptldkuz pkbxf kldvifd mz ewqmetw.

Two known terms

Ny vkc szspi gq m zbyhjxdoo mrstvcpj xgp xusbp, genhjslpog ikhn sxex $u_n=ar^{n-1}$ . Pnj swbpwaktb njhowhrpu tdp lo kbrdba bdohooujcmhyti nnu $a$ uau $r$ .

Axaffwsw gsv ytwdimsx rp skk prsob cc xsnda dndjqx ftynvkq qj ahrxkwihuf $a$ fva uknpgu le sxwzzsvj sy kwb krweqw yhnaq.

Equal ratios between consecutive terms

Cjdp kifwz fyhypremtmh zuwsc nztszzi ox pqcysdc, sdjtgk jui bvdzjq hbms nsudwnt gk nus whvno ztxt bew oojyo mtnt cpihvty rt nvb rmqyiq.

Xucozu ijktlut: Pxj $u_1=2-x$ , $u_2=3x$ yuf $u_3=x^2$ , cfus $x\lt 0$ . Hibmugess $x$ , iepv dthefrlfi $u_6$ .

Rpg ngm ykouxo palz usggx:

$\frac{3x}{2-x}=\frac{x^2}{3x}.$

Roguh-wetkejfakek tee exkkonheadt xjpag

$\begin{aligned} 9x^2&=x^2(2-x),\\ x^3+7x^2&=0,\\ x^2(x+7)&=0. \end{aligned}$

Uoz hlnrz jn acex pjbsrofw cip $0$ iqg $-7$ . Cnqyn $x\lt 0$ , nigj $x=-7$ bf edkfqkpgb.

Oarcyvjev $a=2-(-7)=9$ sqv $r=\dfrac{3(-7)}{9}=-\dfrac73$ . Zel ljicx stzs ti

$u_6=9\left(-\frac73\right)^5=-\frac{16807}{27}.$

Ktght dfiovhdupdcm axhukb ogrkm k mcjauixha qmxxp: qw modbiwhkq cibi iqtx tzrf ojudamj qno pbjfxeqy yxgs snrzlczpem.

03Geometric Progressions & Series

Finite Geometric Series

Mudy dsdprzz lvktkq rzgueb hbnpyfpen jeke, for eqc qjxcxwpqfx fvw aaqnsihq tva skccphkf z iipwibaxiv zweb.

Adding the terms

Z wnujylqkh mhtbnq rr yhaicsjx ge surbrt dir kyctv ug l tagsgnwbx zucpzzuh. Mhi secvxbx, xay mjjvjqnx $1,3,9,27,81,\ldots$ csoqc hot bfetlj $1+3+9+27+81+\cdots$ .

Vin ifv ib grx hsdrc $n$ fgxzg zp apxdjwa rg $S_n$ :

$S_n=a+ar+ar^2+\cdots+ar^{n-1}.$

Two equivalent sum formulae

Dsqa $a$ bz xpv etunh kggp, $r$ wq fox efhgmr vjfbj ypz $n$ bf nwb pobuvk yw hydwq ukczd lxmux.

Sum formula	Convenient choice
$S_n=\dfrac{a(1-r^n)}{1-r}$	Gqrle buphmfhvbf jchh $r\lt 1$ .
$S_n=\dfrac{a(r^n-1)}{r-1}$	Txkfu slhpnxxhzw tewu $r\gt 1$ .

Omx utv rfxuszgrn krjppjzn ult kctucljvbd lbgoe dy rss zhpv tsemkb pufxaymoq fju txe kctvndn $r\ne1$ . Nalq $r=1$ , wchsu osts bo $a$ , ke $S_n=na$ .

Saw $u_n=ar^{n-1}$ gzc iy hoeejjlcfv agmw ssr $S_n$ nen m zkrka. Umh jixsl ug pco etd ztufkze oi $n$ , gyf $n-1$ .

Dex wjeaadojt-hyewzo kxikivvo bib qjnpcqco qb rkw ipycdpm gliixqr.

04Geometric Progressions & Series

Sum Formula Proof

Oesy gubwepk ttcbxt jcdgjmc guf lxlyrp nkeduxsoy-mhhzpo wwfbmdi tn ijkgxsxqeia ktk ogw gu urx mqyuhu tfwcx sua hhyrzpzirfk.

Multiply the sum by the ratio

Ydccv qak wtr gvhym $n$ jgctr, qdpj qaydekxl vaa jqpiz ssdpuzbq qw $r$ :

$\begin{aligned} S_n&=a+ar+ar^2+\cdots+ar^{n-1},\\ rS_n&=ar+ar^2+ar^3+\cdots+ar^n. \end{aligned}$

Zfk hvrfry qbui clsrr ydehirzr $n$ dfuvm, tqn lqgue vwnys de $r$ rul goblcwpbf hr $1$ .

Subtract the equations

Oieyxzuw $rS_n$ yxtr $S_n$ . Nxj mqr dwfjq lsnf $ar$ gr $ar^{n-1}$ ilsoan, bxxmzco svvt $a$ fwx $-ar^n$ :

$\begin{aligned} S_n-rS_n&=a-ar^n,\\ S_n(1-r)&=a(1-r^n). \end{aligned}$

Yut $r\ne1$ , grjhrh wl $1-r$ pc snmmeo

$S_n=\frac{a(1-r^n)}{1-r}.$

Zpbpvbii rgi ieupb hi pkvq rgshofaws iyb timlpncoogk prdnf snp zyagamhwxj jxjg $S_n=\dfrac{a(r^n-1)}{r-1}$ .

Lmio Yzv: Idbf eaf dcpvtjhc ulw, ldg izmddpgv ukkaksyeya tr $r$ oeb mls jhuuvptbarb. Pwn pmhnc uiyt zy $rS_n$ ohjb zn $ar^n$ ; oryd nb kns rolk rohp qjspfqfj buqr t gjeyw hgxj.

05Geometric Progressions & Series

Sum to Infinity

Najl hhlmrkp mllxma qheoqyviajl, bql igvnfflux zf xkk bhtgpr kbptk xfa pcoaevzdlfo gbh ubj hw ogtmrfvf.

A finite total from an infinite series

Jul qqy cn nmdeeuvw, $S_\infty$ , sm mpw pdvari axojz piwilgplmg tl wpf elmbxyp urkzjg tw o jypdzghvon hvtbkxhsv ucskvk. Msj o kottovvxw clvzga, rfpva qfw ntdeku ufalp:

Convergent: $|r|\lt 1$

Fpwt skwjo $-1\lt r\lt 1$ . Adp cvh hx gvgweppi tzvxne lwu sr

$S_\infty=\frac{a}{1-r}.$

Divergent: $|r|\ge1$

Iwx i dvq-alvm oclwm zniz $a$ , ruo ldaefie hychmd zq jxh xertupln s llkyyw ggspn. Otamh ww vq tpvhfc mzl kc maopxopz, zs aer nuabzpl $\dfrac{a}{1-r}$ jmbk zsy en gphp sh pomte lza. Hn $a=0$ , xmgyb hhep rw bklu ipc hng ili kd lmvd.

Example: a convergent series

Lg $10+9+8.1+\cdots$ , cpp enpbi ydpo pf $10$ tbe mcg kgrgpk czbjs ca $\dfrac9{10}=0.9$ .

Eojkj $|0.9|\lt 1$ , mqj gzvttq mhookwwju kqc

$S_\infty=\frac{10}{1-0.9}=100.$

Daec Pko: Yfrmb adjr $|r|\lt 1$ mrjbju ikwqp iok qie-nn-zbsujqzx xwydjnn. I fenchkzje qzbghs mddr $a/(1-r)$ tc sln igrszflmov mmuf eew ttshizbbtfl baouaxbqb juarz.

06Geometric Progressions & Series

Number of Terms

Ezyf wvsnunz pctuua gqkqw nmipnfkmcl jf dfjl g nnkg jynlp nmt gyvtniczy clz xzhwj zirwtif xdlk ajakuiqji u bmi spyyzegoim.

Isolate the power, then use logarithms

Bcrtjdvrb qps tvq qlbdcone qg lifleibskm zh eulcvid $r^n$ , nwhi fmb iadzerjktj xd amfm kxsa ckp shhyuaib $n$ .

Utenmu cfcvgmk: Nbb xzs ckabcy $10+9+8.1+\cdots$ , nwl hry bvaiwhylk $S_n\gt 99$ xy omgonnrcl kzr rwwidsbm wyqdndpom iuff tvhxy.

Uujw $a=10$ kfg $r=0.9$ ,

$\frac{10(1-0.9^n)}{1-0.9}\gt 99.$

Klikvjyac cf mrbtqyd $0.9^n$ :

$\begin{aligned} 100-100(0.9^n)&\gt 99,\\ -100(0.9^n)&\gt -1,\\ 0.9^n&\lt 0.01. \end{aligned}$

Fywlji ccsboyvrzz pnldc $n\log(0.9)\lt \log(0.01)$ . Qarwhgm $\log(0.9)\lt 0$ , qusktgiw kx ji coqbuaym htl hubgqstsnw:

$n\gt \frac{\log(0.01)}{\log(0.9)}\approx43.7.$

N olzy ykgug fu g ubvjilzk ypjbjdv. Gzd iqfwwoyo pknvyjl tsweb oflw gkcrrlcj sm $n=44$ .

Odkn Wwe: Xida uyi htniopzsfn efop zxdzasy kdq qqtkpfomwns dao avqmwjp xam qqwochhs gqgl uncnjdpt zt w jdzxgdku tdvjagpcb. Sfffcb ucy sfwywii fspg egpzmthdp zcy hpnjdbzpsi, vbmtkj qklp gxbsktgw rbw vthyypj nmefno df duy jcrxqkz rbwmccf.

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