Further Exploration of Patterns

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Edmund Poole
6 years ago
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1 Further Exploratio of Patters

2 Abstract Quadratic Patters a+b+c 4a+b+c 9a+3b+c 16a+4b+c 5a+5b+c 1 st chage 3a+b 5a+b 7a+b 9a+b d chage a a a 1 Coefficiet of is ( a) a. Costat secod chage, therefore it is quadratic startig with a. Write out the a series a, 4a, 9a, 16a,... ad subtract from the origial series. a b c, 4a b c, 9a 3 b c, 16a 4 b c, 5a 5 b c, a 4a 9a 16a 5 a, b c b c 3b c 4b c 5 b c, 1 st chage b b b b his is liear b st If you ow go dow form the 1 term which is ( ) b c b c he Geeral erm is ow a b c

3 Geeralisig Quadratic Sequeces 1, 4, 9, 16, 5, 1 st chage d chage, 8, 18, 3, 50, 1 st chage d chage Whe you get half the d chage you get the coefficiet of 4 4, 16, 36, 64, 100,... 1 st chage d chage ,, 4.5, 8, 1.5,... 1 st chage d chage 1 1 1

4 Your ur [.7 pg. 18] Write the Geeral Formula for the followig paters. (a) 3, 1, 7, 48, 75, (b) 0.5, 1,.5, 4, 6.5, Solutios: (a) 3 (b) 0.5

5 More Difficult Quadratic Patters Method 1 6, 1, 0, 30, 4, 1 st chage d chage herefore it is with other terms. Write out the series 1, 4, 9, 16,... ad subtract from the origial series. 6, 1, 0, 30, 4, 1, 4, 9, 16, 5, 5, 8, 11, 14, 17, 1 st chage his is liear 3 st If you ow go dow form the 1 term which is 5 3 he geeral term is ow 3

6 More Difficult Quadratic Patters Method 6, 1, 0, 30, 4,... Look for lowest commo factor, here it is 3, 3 4, 4 5, 5 6, 6 7,... his is a AP AP First AP, 3, 4, 5, 6,.. a ( 1) d ( 1)1 1 Secod AP 3, 4, 5, 6, 7,... a ( 1) d 3 ( 1)1 ( 1) ( ) 3

7 Your ur [.8 pg. 18] Write the Geeral Formula for the followig paters. (a) 5, 1, 1, 3, 45,... (b) 5, 15, 31, 53, 81,... Solutios: (a) + 4 (b)

8 Your ur [.9 pg. 18] 1 st d 3 rd How may blocks are i the 4 th patter? Write a gereral formula to fid the umber of i the th patter. How may blocks are i the 8 th patter?

9 SOLUION 1st d 3rd 4th st chage d chage 4 4 Costat, therefore quadratic

10 Method st chage d chage 4 4 herefore it is with other terms. Write out the series, 8, 18, 3,... ad subtract from the origial series. 4, 1, 4, 40,..., 8, 18, 3,..., 4, 6, 8, 1 st chage his is liear 3 st If you ow go dow form the 1 term which is 0 he geeral term is ow 8 (8) (8) 144 blocks

11 Method 4, 1, 4, 40, Look for lowest commo factor, here it is [Lookig at legthbreadth], 3 4, 4 6, 5 8, his is a AP AP First AP, 3, 4, 5,... a ( 1) d ( 1)1 1 Secod AP 3, 4, 5, 6, 7,... a ( 1) d ( 1) ( 1)( ) 8 (8) (8) 144 blocks

12 Graphig the Couples 8 = 144

13 Give you have 0 metres of wire, what is the maximum rectagular shaped area that you ca eclose? Area st chage d chage herefore it is with other terms. Write out the series 1, 4, 9, 16,... ad subtract from the origial series. 9, 16, 1, 4,... 1, 4, 9, 16,... 10, 0, 30, 40, his is liear 10 he geeral term for the area is 10

14 1 r Geometric ? his is a Geometric Sequece with Each term is multiplied by to get the ext term: (Commo Ratio) a r

15 1 Fidig the of a Geometric Sequece ( ar) r ar 3 ( ar ) r ar 4 5 a ar 3 ( ar ) r ar ar Expoetial

16 A ball is dropped from a height of 8 m. he ball bouces to 80% of its previous height with each bouce. How high (to the earest cm) does the ball bouce o the fifth bouce. 1 st d 3 rd 4 th 5 th 8, 6.4, 5.1, 4.96, 3.768, m = 6 cm Is this a Liear, Quadratic or Cubic patter? Let s look at the graph of this.

17 x Blue: f( x) 8(0.8) or 8(0.8) Write dow the fuctio which describes the red graph. What is the total distace travelled by the ball whe it hits the groud for the 5 th time?

18 What if we were asked to fid the total distace travelled whe the ball hits the groud for the 0 th time. Is there ay geeral way of doig it? S a ( r 1) a (1 ) S r r1 1r

19 S a ar ar ar ar ar rs ar ar ar ar ar ar S rs a ar Subtractig Fidig the S of a Geometric Series (1 r) S a ar S a ar 1 r S a(1 r ) a( r 1) or S 1r r 1 hese formulas ca also be proved by Iductio Lik to Studet s CD

20 A rabbit is 10 metres away from a some food. It hops 5 metres, the hops.5 metres, the 1.5 metres, ad so o, hoppig half its previous hop each time. What will the legth of the 6 th hop be? 5 m.5 m 1.5 m

21 5,.5, 1.5, 0.65, 0.315, What type of patter is this? Discuss

22 A rabbit is 10 metres away from some food. It hops 5 metres, the hops.5 metres, the 1.5 metres, ad so o, hoppig half its previous hop each time. If the rabbit kept hoppig forever, what i theory would be the total distace travelled by it? 5 m.5 m 1.5 m

23 For r 1 r 0 he Sum to Ifiity of a GP a(1 r ) a ar S 1 r 1 r 1 r a 0 S as r 0 for r 1 1r 1r S a 1 r for r 1

24 Extedig the Blocks Questio Fid the total umber of blocks required to make the first 5 patters. 5 5 S r r S S S S r1 r1 ( 1)( 1) ( 1) 6 ( 1)( 1) ( 1) 3 5(5 1)((5) 1) 5(5 1) 3 11,700

25 hree Formulae ( 1) ( 1)( 1) ( 1) r r r 6 3 r1 r1 r1 hese formulas ca be proved by Iductio

26 Summary of GP formulae of a GP 1 ar S a( r 1) a(1 r ) of a GP for r 1 or for r 1 r 1 1r S S 1 S a 1 r for r 1

27 a Express 1. i the form of where a ad b b Method 1 Method his is a ifiite GP with a ad r a S 1 r 10 S Let x 1. 10x 1. x 1. 9x x 9

28 a Express 1.43 i the form of where a ad b b Method 1 Method his is a ifiite GP with a ad r a S 1 r S Let x x x x x 99

29 1 AP AP Show that ( 1) 1 Fill i the various values i the square brackets ad fid the sum to terms of the series whose 1 is ( 1) Fid the sum of the first 0 terms of the series Other ypes of Series 1 S

30 1 3 Solutio S S S

31 Arithmethico Geometric AP x GP 1 r 3r r ( 1) Fid the of the followig sequece, 8, 4, 64, 160 Fid the of the followig sequece 1, 4, 3 8, 4 16, 5 3 Each term i this sequece is a AP GP 1 of AP of GP () Combied, ( )

32 GeoGebra hree Graphs Fuctio Ispector

33 01 LCHL Q4 I a sciece experimet, a quatity Q(t) was observed at various poits i time t. ime is measured i secods from the istat of the first observatio. he table below gives the results. t Q(t) bt Q follows the rule of the form Q( t) Ae, where A ad b are costats. (a) Use ay two of the observatios from the table to fid the value of A ad the value of b, correct to three decimal places. (b) Use a differet observatio from the table to verify your values for A ad b.

SEQUENCES AND SERIES

Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces