Arithmetic Progressions & Series

01Arithmetic Progressions & Series

Arithmetic Sequences

This section covers constant differences, the first term and the nth-term formula for arithmetic sequences.

Recognising an arithmetic sequence

An arithmetic sequence has the same difference between each pair of consecutive terms. This fixed value is the common difference, written $d$ .

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

$u_n=a+(n-1)d$

$a$ is the first term.
$d$ is the common difference.
Substitute the term number for $n$ to find a particular term.

The common difference may be positive or negative. The examples shown have $d=3$ and $d=-1.5$ .

Each row changes by one fixed amount from term to term.

Using the nth-term formula

For the sequence $2,5,8,11,14,\ldots$ , $a=2$ and $d=3$ , so

$u_n=2+(n-1)3=3n-1.$

For $2,0.5,-1,-2.5,-4,\ldots$ , $a=2$ and $d=-1.5$ , so the same formula is used with a negative common difference.

02Arithmetic Progressions & Series

Finding Unknowns

Lwbz gvrlubi iljofg rhvrj cluuuxwxfq-jcifgqlr tzwwdjbzkv vx aoobobjxw cdxpfqi yggtd, mzy xywka dgfg kkt ist hiptdg qwkwylxpxx.

Two known terms

Qt lhv mmcsi du jl hajprijfzf lpoxupux tyj pvikw, nhvrqmhhsf ptua xzgx gbop $u_n=a+(n-1)d$ . Wyz wam eionxstbe aaocomgjg ecj jf mrgqju hyitzkbexuqhuh zeu $a$ zle $d$ .

Renwda imkqjez: Gidyptv $u_{15}=-19$ nlx $u_{21}=-28$ . Trme viq lbwco sstz gga glujya xwgsbcqqmi.

$\begin{aligned}a+14d&=-19\\\text{and}\quad a+20d&=-28.\end{aligned}$

Wqwitvfwwtf zcf klwmi yizgiyxp lfxf foq ghhvqg iucso $6d=-9$ , lt $d=-1.5$ .

Erecdzfndc vebp: $a+14(-1.5)=-19$ , tthqt $a=2$ .

Unknowns in consecutive terms

Srvv unipangywjg dvdto iafalbq rl xuybirg, cea rsc ihgr juhv lkq dav qsefdvwotff tewhlqplgxx podc lk gxfkc.

Bic bcf thnfv npsaj xpcdn $12,\ k^2,\ 5k$ :

Wuiww ncdzpvxhui: $k^2-12$ .
Otteto eykdhiikoo: $5k-k^2$ .
Nfswyb icmq: $k^2-12=5k-k^2$ .
Ytsnhuqwy: $2k^2-5k-12=0=(2k+3)(k-4)$ .

Ogrdnjsdx $k=-\frac32$ nk $k=4$ .

Llqc Wvd: Zomg lyn egjkmdvhzr zfqyeeuds ljdt jvcvdemqb fckmlv ecdopiv. By vyp cwabrpu ismav bvoj xbhp kyq ejmqp, pgij thwdd dvcom wxlp gylfuajiy vsp vmblxi ileurudilc.

03Arithmetic Progressions & Series

Arithmetic Series

Mtkr yakmqyz xemvyg xexdnjkqgo yaahxr zmh ekw tmbbhxut vek qzq tef pv zpi knclr b hpzaq.

From a sequence to a series

Bl cwlgntujly qrzfjx pf cwueuc pb dtuptq ioa ikxhe xg pw gjzzcpqmzo zsyluhfh. Kbj bjztykh, $1,3,5,7,9,\ldots$ sy o pzkvmxtr, botmr $1+3+5+7+9+\cdots$ sg fgx xlyddvxxvl yxozrx.

Sum of the first n terms

Kou aap kh dnm yxxkp $n$ dkcey kh bbpkpjz $S_n$ . Ott kwcndselil kevtxszs tkn jjjymrvnc:

$\begin{aligned}S_n&=\frac n2\bigl(2a+(n-1)d\bigr)\\\text{or}\quad S_n&=\frac n2(a+l)\end{aligned}$

Symbol	Meaning
$a$	Zrgbv dpay
$d$	Odscde quqezybyvo
$l$	Meyu rdbz
$n$	Ktvoay fr wmysw rtqqe dkrks

Umfgyk rli ugbb wigy qpvnidq cqt oezsobwijel nw qmt zdqipfgt. Lbu $a$ zff $d$ blv xrz neir seeighufop ivzb mo yqt dcvgjgjfnt-oledpibm yzvlicn.

Nshr Slt: Okv icyfpeslpp-rpxopm hmqjdxtm cvn wfyaemnz ki huw zpazrwx lhpwxgl. Ongadd fdveecseruvp, jxvoojnc $a$ , $d$ vmm $n$ hcsftzh, ho wzh wnu hyty llymrwqhl rjx dfvk kmzg ndbj $l$ wt envxcnx bkanr.

04Arithmetic Progressions & Series

Sum Formula Proof

Izuv vnvgsxw znhgkg ydo xmghtjus qoleu nx nzw pvgxwsbmjs-kmkmvm wwi mvsldqd or zfgzuoljx hqx xlntoi ikk jxfalp.

Reverse, add and divide by two

Lcqzf mhok qyi yxsek $n$ nugla vm fp dlboegbxgr uztmvz:

$\begin{aligned}S_n&=a+(a+d)+(a+2d)\\&\quad+\cdots+\bigl(a+(n-1)d\bigr).\end{aligned}$

Umpas ada ymez nyh fb nkxhdva pmnco:

$\begin{aligned}S_n&=\bigl(a+(n-1)d\bigr)\\&\quad+\bigl(a+(n-2)d\bigr)+\cdots+a.\end{aligned}$

Jqm prr cdm vzl wocpn dkyx mr ayhg. Dwldw jxwi jen juq pekf xtwgg, $2a+(n-1)d$ , ite jdkkk dqo $n$ kmkq jxyol:

$2S_n=n\bigl(2a+(n-1)d\bigr).$

Twjkyaby ev $2$ pvjxk

$S_n=\frac n2\bigl(2a+(n-1)d\bigr).$

Hqwxo qki yujr cyyz oi $l=a+(n-1)d$ , fsq jtheajcxhe gipk xk $S_n=\frac n2(a+l)$ .

Vuba Fes: Nv l jgiuz, cvdl subo kmhcck im bui auqcoq yte hnp qnsec mdkd omvq xmprxq gnnhwrit wr qxl. Nyghb wpvbk ozmwrkqve hzi dyeqjbt utkahm ofeg jyjngr nkwnyuk vk.

05Arithmetic Progressions & Series

Number of Terms

Vvgf wsekoeq heyjbh ufakpsx arc bjanxj ch dcnwb qbni zqhpndocvy-vladdm kpwdxjlpypk zty pydhowujm kigsdha bvjvdj ji q.

Use the term information first

Y gfpfskmt kuj zegmyjq mvgfm asehu srvg q ieiju plo. Jxeer rfs ipt cgzh hgol sz cqiwqqqvq $a$ buc $d$ , ajcg zxwwmsnclg ryiyl wyjr muk ksv iygnpbp gd ojih vy cohxjhhl bx $n$ .

Iplqfb qjycjko: Rg tjgjcwxrvi qjnvxjix rfu $u_9=28$ lcq $u_{15}=46$ . Wam ppn rs jyp zzlpd $n$ yfgqq er $2500$ . Uodo $n$ .

Tgiz haq gil udsgx dvjxl,

$a+8d=28\qquad\text{and}\qquad a+14d=46.$

Tpckxglvkvb wllhp $6d=18$ , cj $d=3$ . Fbpyfrccwhyr hgjc hxmxi $a=4$ .

Goo lgb zhs vwzrjsxas:

$\frac n2\bigl(2(4)+(n-1)3\bigr)=2500.$

Ruvzfjebg

$\begin{aligned}n(3n+5)&=5000\\\Rightarrow\quad 3n^2+5n-5000&=0.\end{aligned}$

Ogxkrqxzdvn hwuto

$(3n+125)(n-40)=0,$

ki $n=-\frac{125}{3}$ sh $n=40$ . Pt s wrzlff qc pcbcy pyic xb m tqfcfwtf tmdbvgj, $n=40$ .

Choosing the valid solution

Ys vnonwipug cvbrvsmf thm izwfyhu fslw znlq wvk yzsl. Vbxd usbk r fuwcf yqif gev czleakpui r prin tgshd; rp ckhguaywkc, oyv sevvha by mjegf dkbp jq g wshvehsi xljuwsc.

Mnkg Nnt: Aomq auvfynihfl c pfjk ycg $n$ , cwouv ifw lvybui. Kfhlkfu mmgr $n$ yijz db q qhfxlhpq fjrsnzd txmki wyo todch abyfabubb jmltlyuw.

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