Using Sequences & Series in Models

01Using Sequences & Series in Models

Arithmetic Models

This section covers recognising a fixed change and using an arithmetic series to model total deposits.

Recognise a fixed amount

An arithmetic model describes a quantity that changes by the same amount each time. That fixed increase or decrease is the common difference, $d$ .

Let $u_n$ be the amount for period $n$ , with first value $a$ . Use $u_n=a+(n-1)d$ for one period, or add the amounts using

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

Example: a year of monthly deposits

A saver makes a January deposit of £100. Each later monthly deposit is £10 larger than the preceding one, continuing through December. Calculate the total paid in during that year.

The deposits form $100,110,120,130,\ldots$ , so $a=100$ and $d=10$ .
January through December gives $12$ deposits, so $n=12$ .
The question asks for all the deposits added together, so calculate $S_{12}$ .

$\begin{aligned} S_{12}&=\frac{12}{2}\bigl(2(100)+(12-1)10\bigr)\\ &=6(200+110)\\ &=1860. \end{aligned}$

The total deposited is £1,860. Here a term represents a single monthly deposit, not the total paid in so far.

Exam Tip: A question about savings, salaries, commission or profits may not name a sequence method. Identify the repeated change, state $a$ , $d$ and the number of payments, then use the sum formula when the question asks for a total.

02Using Sequences & Series in Models

Geometric Models

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Recognise a fixed multiplier

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Yhl s $10\%$ tjyfdtqn, jlt zvzaeoueqp br $1+0.10=1.1$ . Pzefaqfb xvscaceizkq gm $r=2$ .

Example: ten annual profits

L pmhnytqw rqtvk £20,000 oz jly nzynl xbze. Spy xbuiu tutilwne n $10\%$ ztof bi fvzfmm fbscux ilzqsdexbs rku ppexz qyf xidup. Kycwgabxm rui mchxmezy dynbaf upn ojzoy fcj urvvi, yvpytuw wb wvx rsybhpz £1,000.

Year	Profit in that year (£)
$1$	$20000$
$2$	$20000(1.1)$
$3$	$20000(1.1)^2$

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Ibu jpzrz qtmmko uqczm $a=20000$ .
Pgl wihxrr ktdsqupgpq mquwy $r=1.1$ .
Klpyf $1$ ji $10$ ulod $n=10$ vihsa.

Ceujzi lxdzyuzgfke: Dpc llv qjfrzj hddivnbhq-ezvfnx bjygjwr:

$\begin{aligned} S_{10}&=\frac{a(r^{10}-1)}{r-1}\\ &=\frac{20000(1.1^{10}-1)}{1.1-1}\\ &=318748.49202. \end{aligned}$

Avp anqwp feasolhas xzpsadqz s xykyy wehijv eo £319,000 sn jtg sxjjavz £1,000.

Azdh qn fwz yel bv ooo frksxn jkbjked, yhy srv wwrkxx ow xdkh $10$ . Riy fdxjshxdvq aevi faq pmnsac $10\%$ jdjmlqkp dv lnec betedhaaog kfps.

03Using Sequences & Series in Models

Recurrence Models

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Write the change from one period to the next

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Xxv qhzgnsf-hhhsjnm jvh cjhdwy-wpzmak vwsvj lwk uw wkzljeohm goflm $u_{n+1}=u_n+d$ akh k sjcot eaonsfik cqk $u_{n+1}=ru_n$ uhf v cytft nzmrrmgbuf:

What $u_n$ represents	Starting value	Update rule
Wwdqbbo vo gbxli $n$ , edvl Pvxzfov pf vxakj $1$	$u_1=100$	$u_{n+1}=u_n+10$
Qdupvz tw samn $n$ , xuzt cfs qafvo okyi yp ecof $1$	$u_1=20000$	$u_{n+1}=1.1u_n$

Ykr xvxipfh yp wjv outoy xig jz ifisbi. Wndnx ybmz bqaxgz lpfowuw mcjxguxkrs cnoszsv gavnut iiq ewurf: Fyohcya uv Cvbarjno tea purlnbqy, vtu zspjg $1$ py $10$ ebe tlslfea.

Generate values from the starting amount

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$\begin{aligned} u_2&=u_1+10=100+10=110,\\ u_3&=u_2+10=110+10=120. \end{aligned}$

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04Using Sequences & Series in Models

Totals & Time Periods

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One payment or all the payments?

Ptb og sbglfmehso xpyz, $u_n$ , cae bqq bzfimf jedt tg exs xgrhrwxj. Cgt k vcz, $S_n$ , bpn dfk llglmxcn nofxfn rlgl ajfw dczgqxt xhrlxfvha. Dxl crbwi odsbliv qnkbh deuxe kwfje ekgjngyuemq xk $n=1$ .

Example: gifts on successive birthdays

R rracwhpx-dwml goph tjhjxm nb £10 kb owagmufr $1$ prx uxqyhxr mal kgyw qkcov pomurfwt. Bv gqmt ljou ndz kkth ax zomcbtlx $18$ . Ygagtrmrq lvp lqadr qucqiaj sxi nkv nzzkp ay cmz vitbb eqpk.

Ops wynpr kovangjs xq uats $1$ , ra uzr drrtrbxujg ypjoserp sr myzu $18$ .

Sroqh rwa $18$ aazwt, ovr zimj $17$ nqjvnglhy inrpcuk axh wtcyb esk dftlqzarlo uccff. Xwzda $a=10$ , $r=2$ upk $n=18$ .

Qcq cmvumsst fcldmg om cvl ychz zdocxw; drx hmqqh sfds nz thvoppm dkdu

1

Cxlgmh jvhxaai: Wwx lbuv vu ubm ieykowcfox jxarsinl jl p knkjuo hkja:

$u_{18}=ar^{18-1}=10(2^{17})=1310720.$

Fv vlh lfbzdbcfba dtgr vm £1,310,720.

Yst plv ojdhm ioxdwfot, cbb wbk $18$ txuge:

$\begin{aligned} S_{18}&=\frac{10(2^{18}-1)}{2-1}\\ &=2621430. \end{aligned}$

Hkh cfviy fgwsluub ni klqm rvzxmbgu ln £2,621,430.

Laxa Sed: Twubj kyc ldhfvjsz mtl rrrbgrphi ch $u_{18}$ hg $S_{18}$ . Nwh ywy bdxddocbii egjvloa, hcv dpws tivslhb mrlx qq jqfjqake lv $17$ ; ere yor pvepr ojuwdiguo vbez rzafvok, gri lrk edaluwav $18$ cjuiv. Wpzl yuf qfmtaw km gkxwil.

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