Sequences and Series 4

Transcription

1 Sequeces ad Series 4 LEARNING OBJECTIVES I this chapter, you lear how to use GC to Fid the th term of a Sequece. Fid the Sum of a Sequece. Fid the Sum to Ifiity of a GP. Solve Quadratic Equatio uder EQUA mode. Evaluate Σ. Fid the Root(s) of a Equatio uder GRAPH mode. Determie the Behaviour of a Sequece. Example 4. A geometric series has commo ratio r, ad a arithmetic series has first term a ad commo differece d, where a ad d are o-zero. The first three terms of the geometric series are equal to the first, sixth ad teth terms respectively of the arithmetic series. (i) Show that 5r 9r (ii) Deduce that the geometric series is coverget ad fid, i terms of a, the sum to ifiity. (iii) The sum of the first terms of the geometric series is deoted by S. Give that a > 0, fid the least value of for which S exceeds 99% of the sum to ifiity. 8 GLM PTE LTD

2 So l u t i o a+ (6 ) d a+ (0 ) d (i) r a a+ (6 ) d d + 9 d a r + 5 a d + 5 a r + 9 d r 5 r r r + 9 5r 5 9r 9 a 5 r r r (show) If x, y ad z are i G.P., y z r x y (ii) 5r 9r+ 4 0 ( r )(5r 4) 0 r or r 5 Sice d 0, the three terms of GP are ot the same, thus r. 4 4 Hece, r Sice r <, the geometric series is coverget. 5 a Ad the sum to ifiity 5 a. 4 5 (iii) S > 0.99S a( 0.8 ) a > > < 0.0 lg 0.8 < lg 0.0 lg 0.0 > lg0.8 > 0.6 Hece, the least value of is. UNIT 4 Sequeces ad Series (You may use GC to verify your aswer.) GLM PTE LTD Practical Guide To GCE A Level H Mathematics GC Approach 9

3 p iq uuq uy 0.8^ f$,f,, 00, kl Select RUN.MAT mode. Select LIST. Select SUM. Select Seq. Eter the parameters i the followig format: Seq (Expressio, Variable, Start, Ed, Icremet) Aswer: 4 which is S where a 0.8 ad r 0.8. (Here, we assume a takes a positive value.) p8 eq 0.8^ fl Select RECUR mode. Select TYPE, F: a A + B. Eter the expressio 0.8. y l50 ll Lp ql Select SET. Eter for Start, 50 for Ed. Icrease the Ed value if foud ecessary later. Eter SET UP. Set O for Display. 30 GLM PTE LTD

4 u N Select TABL. Press N util a exceeds 0.99S, i.e You may also cosider 0.8 fallig below 0.0. Both approaches arrive at the same aswer: the least value of is. Example 4. (i) Patrick saves $0 o Jauary 008. O the first day of each subsequet moth he saves $4 more tha i the previous moth, so that he saves $4 o February 008, $8 o March 009, ad so o. O what date will he first have saved over $5000 i total? (ii) Key puts $0 o Jauary 008 ito a bak accout which pays compoud iterest at a rate of 3% per moth o the last day of each moth. He puts a further $0 ito the accout o the first day of each subsequet moth. (a) How much compoud iterest has his origial $0 eared at the ed of 3 years? (b) How much i total, correct to the earest dollar, is i the accout at the ed of 3 years? (c) After how may complete moths will the total i the accout first exceed $5000? UNIT 4 Sequeces ad Series GLM PTE LTD Practical Guide To GCE A Level H Mathematics GC Approach 3

5 So l u t i o (i) T 0, d 4 [ (0) + 4( ) ] > > 0 < 54.7 or > 45.7 Hece, Patrick will first have saved over $5000 i total o October 0. (You may use GC to fid the roots.) pf w q l9l 50 0l q Select EQUA mode. Select F:Polyomial. Select Degree. Eter the values of a, b ad c. Select SOLV. Aswers: 45.7 or 54.7 (ii) (a) The required compoud iterest 0(.03) 36 0 $38.0 (3 s.f.) (b) The required amout 0(.03) + 0(.03) (.03) (.03) (.03) $303 (You may use GC to verify your aswer.) p8 e w Select RECUR mode. Select TYPE. Select F: a + Aa + B + C. 3 GLM PTE LTD

6 .03 jw+ 0kl Eter the recurrece formula. Press w for a. y N00 l Select SET. Eter the start ad ed value of as well as the value of a. du Select TABL. N Keep pressig N util the value o the left colum reaches 36. Record the correspodig value o the right colum: (c) 0(.03) 0(.03)... 0(.03) > 0(.03)(.03 ).03 > > 8.8 lg8.8 > lg.03 > 7.5 The total i the accout will first exceed $5000 after 7 moths. (You may use GC to verify your aswer.) UNIT 4 Sequeces ad Series GLM PTE LTD Practical Guide To GCE A Level H Mathematics GC Approach 33

7 N Keep pressig N util the value o the right colum first reaches 5000 or above. Record the correspodig value o the right colum: 7. Example 4.3 A sequece u, u, u 3, is such that u ad , for all. u u ( + ) (i) Use the method of mathematical iductio to prove that u 3. (ii) N Hece fid 3 3. ( + ) (iii) Give a reaso why the series i part (ii) is coverget ad state the sum to ifiity. (iv) Use your aswer to part (ii) to fid N ( ) 34 GLM PTE LTD

8 So l u t i o (i) (ii) Let P be the statemet u where u u ad u for. ( + ) Whe, L.H.S. u ; R.H.S.. L.H.S. R.H.S. P is true. + Assume P k is true for some k, i.e. uk. 3 k To prove P k+ is true, i.e. uk+ 3 ( k + ) L.H.S. u k+ 3k + 3k + k+ uk 3 3 k ( k + ) 3 3 k + k k k ( k + ) 3 ( ) 3 3 k + k k 3 3 k ( k + ) k ( k + ) 3 k k k k k ( k + ) 3 R.H.S. P is true wheever P is true. k Sice P is true ad P P is true by Mathematical Iductio, + P is true for all. N k 3 N N + k+ N + + ( u u+ ) ( + ) u u + u u... + u u + u u N u u N N+ ( N + ) 3 GLM PTE LTD Practical Guide To GCE A Level H Mathematics GC Approach 35 UNIT 4 Sequeces ad Series

9 (You may use GC to verify your aswer.) p ruw N!f$ B8 N!!!!z3f s+3f +Nf ^3$j f+k ^3l -z N9^3 l Select RUN.MAT mode. Select MATH, (. Eter the iitial value ad the ed value 8 (radomly chose). Eter the expressio. Aswer: 78/79. If the aswer is ot i fractio form, press x. Evaluate the aswer i part (ii) whe N 8. Aswer: 78/79 (iii) N Whe N, 0 ( N + ) ( N + ) ( + ) Hece, the series i part (ii) is coverget ad the sum to ifiity is. 36 GLM PTE LTD

10 (iv) N N ( r+ ) 3( r+ ) whe r+ ( ) r ( r+ ) r N 3( r+ ) 3( r+ ) + r 3 3 N 3r + 6r+ 3 3r 3+ r 3 3 N 3r + 3r+ r r ( r+ ) r ( r+ ) ( N + ) N r ( r+ ) 3 (You may use GC agai to verify your aswer.) [Apply similar keystrokes foud i part (ii).] Aswer: 5/5 Aswer: 5/5 UNIT 4 Sequeces ad Series GLM PTE LTD Practical Guide To GCE A Level H Mathematics GC Approach 37

11 Example 4.4 x The diagram shows the graph of y x e. The two roots of the equatio are deoted by α ad β. where α < β. (i) Fid the values of α ad β, each correct to 3 decimal places. A sequece of real umbers x, x, x 3, satisfies the recurrece relatio for. x + e x (ii) Prove algebraically that, if the sequece coverges, the it coverges to either α or β. (iii) Use a calculator to determie the behaviours of the sequece for each of the cases x 0, x 3, x 6. (iv) By cosiderig x + x, prove that x+ < x if < x <, x > x if x < or x >, + (v) State briefly how the results i part (iv) relate to the behaviours determied i part (iii). 38 GLM PTE LTD

12 So l u t i o p5 f-l GjfM kl Eter GRAPH mode. Eter the equatio of the graph as Y. u Select DRAW. yq Fid the first root α. Aswer: 0.75 $ Fid the secod root β. Aswer: UNIT 4 Sequeces ad Series GLM PTE LTD Practical Guide To GCE A Level H Mathematics GC Approach 39

13 (i) From GC, α 0.75 ad β (ii) x+ e x x x + e x x + e 0 L L e 0 give that x L ad x+ L whe. x Sice α ad β are the roots of x e 0, hece x coverges to α or β if the sequece coverges. (iii) p8 e w 0.5L GjwM kl Eter RECUR mode. Select TYPE. Select F: a + Aa + B + C. Eter the recurrece equatio. w correspods to a. y w l0 l0l du Eter Table Settig. Chage a 0 to a. Apply the settigs as show. Use a larger value for Ed if later foud isufficiet. Use a 0 for the first case. Select TABL. 40 GLM PTE LTD

14 N While pressig N, observe the chage i a +. It is oted that the sequece coverges to which is α whe x 0. Repeat the same process for the cases x 3 ad x 6. [Set a 3 ad apply similar key strokes.] It is oted that the sequece coverges to also which is α whe x 3. UNIT 4 Sequeces ad Series GLM PTE LTD Practical Guide To GCE A Level H Mathematics GC Approach 4

15 [Set a 6 ad apply similar key strokes.] It is oted that the sequece diverges whe x 6 as show by the 'ERROR' message. x x x (iv) x+ x e x e x ( x e ) x x If < x <, x e > 0 ( x e ) < 0 x+ x < 0 x+ < x. x x If x < or x >, x e < 0 ( x e ) > 0 x+ x > 0 x+ > x. x (v) For x 0 where x< x+ e x, hece < e x< x < x3 <... <. x For x 3 where < x< x > x+ e > e, hece x > x > x3 >... >. If x 6 where x > x < x, hece < x < x < x GLM PTE LTD

16 Checklist GC Techiques covered i this chapter TECHNIQUES Fid the th term of a Sequece. Fid the Sum of a Sequece. Fid the Sum to Ifiity of a GP. Solve Quadratic Equatio uder EQUA mode. Evaluate Σ. Fid the Root(s) of a Equatio uder GRAPH mode. Determie the Behaviour of a Sequece. UNIT 4 Sequeces ad Series GLM PTE LTD Practical Guide To GCE A Level H Mathematics GC Approach 43