Recurrence Relations. Each correct answer in this section is worth two marks.
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1 Recurrence Relations Paper1SectionA Each correct answer in this section is worth two marks. 1.Asequenceisdefinedbytherecurrencerelationu n+1 =2u n +3andu 0 =1. Whatisthevalueofu 2? A. 7 B. 10 C. 13 D. 16 C 1.4 C 0 0 CN A P1 Q2 2.Asequenceisdefinedbytherecurrencerelationu n+1 =3u n +4,withu 0 =1. Findthevalueofu 2. A. 7 B. 10 C. 25 D. 35 C 1.4 C 0 0 NC A P1 Q1 hsn.uk.net Page 1
2 3.Asequenceisdefinedbyu n+1 =3u n +4withu 1 =2. Whatisthevalueofu 3? A. 34 B. 21 C. 18 D. 13 A 1.4 C 0 0 NC A P1 Q1 4.Asequenceisdefinedbytherecurrencerelationu n+1 = 3u n +7withu 0 =2. Whatisthevalueofu 2? A. 1 B. 1 C. 4 D. 10 C 1.4 C NC A11 HSN 157 hsn.uk.net Page 2
3 5.Asequenceisdefinedbytherecurrencerelationu n+1 = 1 4 u n +8withu 0 =32. Evaluateu 2. A. 10 B. 12 C. 16 D. 32 B 1.4 C NC A11 HSN Asequenceisgeneratedbytherecurrencerelationu n+1 = 1 4 u n +7,with u 0 = 2. Whatisthelimitofthissequenceasn? A B C. 3 D. 28 C 1.4 C 0 0 NC A P1 Q7 hsn.uk.net Page 3
4 7.Asequenceisgeneratedbytherecurrencerelationu n+1 =0 7u n +10. Whatisthelimitofthissequenceasn? A B C. D A 1.4 C 0 0 NC A P1 Q6 hsn.uk.net Page 4
5 8.Asequenceisdefinedbytherecurrencerelationu n+1 = 2 5 u n +6withu 0 = 10. Whatisthelimitofthesequence? A. 10 B. 2 5 C D. 30 A 1.4 C NC A13 HSN 088 hsn.uk.net Page 5
6 9.Asequenceisdefinedbytherecurrencerelationu n+1 = 1 9 u n 2,withu 0 =5. Whatisthelimitofthesequence? A. 9 4 B C D. 9 4 A 1.4 C NC A13 HSN 146 hsn.uk.net Page 6
7 10.Asequenceisdefinedbytherecurrencerelationu n+1 = 1 7 u n +6withu 0 =0. Whatisthelimitofthesequence? A. 7 B C. 1 7 D A 1.4 C NC A13 HSN 167 hsn.uk.net Page 7
8 11.Asequenceisdefinedbytherecurrencerelationu n+1 =au n + 3 2,withu 0 =5. Giventhatthissequencehaslimit1,whatisthevalueofa? A. 1 2 B. 1 3 C. 1 3 D. 1 2 A 1.4 C NC A13, A14 HSN 141 [ENDOFPAPER1SECTIONA] hsn.uk.net Page 8
9 Paper1SectionB 12.Asequencesisdefinedbyu n+1 =au n +bwhereaandbareconstants. Giventhatu 0 =3,u 1 =8andu 2 =33,findthevaluesofaandb. 4 4 C CN A11,A14 a =5,b = 7 Ex ss: formequationinaandb 2 ss: formsecondequation 3 pd: processfora 4 pd: processforb 1 8 =3a +b 2 33 =8a +b 3 a =5 4 b = 7 [ENDOFPAPER1SECTIONB] hsn.uk.net Page 9
10 Paper 2 1.Duringheavyrainfallthedepthofariverisincreasingby2 5unitseveryhour. Anti-floodmeasuresreducethedepthoftheriverby10%eachhour. Theinitial depth is 13 2 units. (a) (i)writedownarecurrencerelationforu n,thedepthoftheriver,inunits, after n hours. (ii)hencefindthedepthoftheriveraftertwohours. 3 (b)ithasbeendecidedthatifthedepthoftheriverexceeds29unitsthennearby houses must be evacuated. Will the houses have to be evacuated? Explain your answer. 3 (a) 3 C CN A10,A11 (i)u n =0 9u n , Ex (ii)15 4units(1d.p.) (b) 3 C CN A12, A13 no need to evacuate 1 ic: interpretdecayfactor 2 ic: staterecurrencerelation 3 ss: evaluatesuccessiveterms 4 ss: strategyforlimit 5 pd: process 6 ic: interpretresult statedorimpliedby 2 2 u n =0 9u n ,u 0 = u 1 =14 38,u 2 =15 4 sothedepthis15 4units(1d.p.) 4 e.g.l=0 9l +2 5orl = <0 9 <1sol =25units 6 willnotexceed29units,sononeed to evacuate hsn.uk.net Page 10
![[SQA] 2.](/docs-images/74/70131638/images/11-0.jpg "(a) 5 C CR A10, A13 1989 P2 Q6 (b) 1")
11 [SQA] 2. (a) 5 C CR A10, A P2 Q6 (b) 1 C CR CGD hsn.uk.net Page 11
12 3.Asequenceisdefinedbytherecurrencerelationu n =0 7u n 1 +3,withu 1 =5. (a)calculatethevalueofu 2. 1 (b)whatisthesmallestvalueofnforwhichu n >9? 2 (c) (i)statewhythesequencehasalimit. (ii) Find the limit of the sequence. 3 (a) 1 C CN A11 u 2 =6 5 WCHSU1Q9 (b) 2 C CR A11 n =6 (ci) 1 C CN A12 since 1 <0 7 <1 (cii) 2 C CN A13 L = 10 1 pd: calculateterm 2 ss: evaluatesuccessiveterms 3 ic: statevalueofn 4 ic: statelimitcondition 5 ss: knowhowtofindlimit 6 pd: processlimit 1 u 2 = =6 5 2 u 3 =7 55,etc.leadingtou son =6 4 since 1 <0 7 <1 5 L =0 7L +3orl = 6 L =10 b 1 a [SQA] 4. Asequenceisdefinedbytherecurrencerelationu n+1 =0 3u n +5withfirstterm u 1. (a)explainwhythissequencehasalimitasntendstoinfinity. 1 (b)findtheexactvalueofthislimit. 2 (a) 1 C CN A P1 Q11 (b) 2 C CN A13 hsn.uk.net Page 12
13 5.A sequence is defined by the recurrence relation u n+1 = au n +b, where 1 <a<1andu 0 =8. (a)giventhatu 1 =12andu 2 =13,findthevaluesofaandb. 3 (b)findthelimitofthesequenceasn. 2 (a) 3 C CN A14 a = 1 4,b =10 OB (b) 2 C CN A13 l = 40 3 = ss: formtwolinearequations 2 pd: eliminateonevariable 3 pd: complete 4 ss: knowhowtofindlimit 5 pd: processlimit 1 8a +b =12, 12a +b =13 2 4a =1 3 a = 1 4,b =10 4 l = l = 40 3 =131 3 hsn.uk.net Page 13
14 [SQA] 6. (a) 3 C CR A P2 Q3 (b) 5 C CR A14, A13 hsn.uk.net Page 14
15 [SQA] 7. Trees are sprayed weekly with the pesticide, KILLPEST, whose manufacturers claimitwilldestroy65%ofallpests.betweentheweeklysprayingsitisestimated that500newpestsinvadethetrees. A new pesticide, PESTKILL, comes onto the market. The manufacturers claim that itwilldestroy85%ofexistingpestsbutitisestimatedthat650newpestsperweek will invade the trees. Which pesticide will be more effective in the long term? 7 7 C CR A14, A12, A13 PESTKILL 1995 P2 Q3 [ENDOFPAPER2] hsn.uk.net Page 15
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