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My Add Maths Modules Form 5 - Progressions

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My Add Maths Modules Form 5 - Progressions Powered By Docstoc
					My Additional Mathematics Modules
Form 5
(Version 2007)

Topic: 12

by

NgKL
(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.&Lship,Cert.NPQH.)

12.1 ARITHMETIC PROGRESSIONS (A.P.)
IMPORTANT POINTS: 1. A SEQUENCE is a set of terms which are written in a definite order, obeying a certain rule. Example: 2, 4, 6, 8 ……. ( a sequence of even numbers) A SERIES is the sum of the terms of a sequence. Example: 2 + 4 + 6 + 8 + ……. ARITHMETIC PROGRESSIONS (A.P.): * * * is a number sequence where the difference between each term (after the first term) and the preceding term is constant. the constant is known as the common difference, d. d = Tn+1 - Tn where Tn+1 = the (n +1) term, Tn = the nth term.

2.

3.

*

Steps to determine whether a given number sequence is an A.P.; (i) Take any three consecutive terms in a given number sequence; Tn, T n+1, T n+2. (ii) Calculate the values of Tn+2 - T n+1 and Tn+1 - Tn (iii) If Tn+2 -T n+1 = Tn+1 - Tn, then the number sequence is an A.P. (iv) If Tn+2 -T n+1 ≠ Tn+1 - Tn, then the number sequence is not an arithmetic progression. The nth term, Tn of an A.P.; Tn = a + (n – 1)d where a = first term, d = common difference.

*

*

The sum of the first n terms, Sn, of an A.P.; Sn = n/2{2a + (n – 1)d} or Sn = n/2{a + ℓ} where a = first term, ℓ = last term. where a = first term, d = common difference.

*

The nth term, Tn, of an A.P. can also be determined by the following formula; Tn = S n
─

Sn – 1

Exercise 12.1 1. Determine whether the following number sequences are A.P. If so, state its common difference. (c) 7/8, 27/40, 19/40, 11/40 …..

(a) 11, 15, 19, 23, …..

(b) 0.2, 0.6, 1.0, 1.4 …..

(d) x, 3x, 5x, 7x, …..

2.

Determine the nth term, Tn, for each of the following A.P. (c) 8, 9¼, 10½, 11¾, ….. T12

(a) 30, 23, 16, 9, ….. T10

(b) 16, 22, 28, 34, ….. T15

(d) -10, -4, 2, 8, ….. T8

3.

Determine the number of terms, n of each of the following A.P. (c) 4⅓, 3⅔, 3, ….., -10⅓.

(a) 36, 43, 50, ….., 323.

(b) 0.2, 0.6, 1.0, ….., 5.4.

(d) -10, -4, 2, ….., 212.

4.

Calculate the sum of all the terms of each of the following A.P. (c) 8, 9¼, 10½, ….. till the 16th term.

(a) 18, 10, 2, ….. till the 12th term.

(b) 12, 15, 18, ….. till the nth term.

d) -13, -5, 3, ….. till the 20th term.

5.

Calculate the number of terms and the sum of all the terms of each of the following A.P. (d) 56, 52,48, ….., -88.

(a) 15, 24, 33, ….., 186

(b) 6.2, 5.6, 5, ….., -18.4.

(e) 48, 55, 62, …., 265.

(c) 7, 11, 15, ….., 163.

(e) 273, 286, 299, ….., 416.

6.

Calculate the sum at a certain nth term of each of the following A.P. (c) 32.4, 33.6, 34.8, …..; The sum from T20 to 27.

(a) 17, 21, 25, …..; The sum from T10 to T19.

(b) 16, 28, 40, …..; The sum from T5 to T15.

(d) 4, -2, -8, …..; The sum from T8 to T20.

(c) 11, 7, 3, …..; The sum from T7 to T25

(f) ¼, 1½, 2¾, …..; The sum from T3 to T10.

Exercise 12.2 – Problem Solving I 1. Given the sum of the A.P. 13, 21, 29, … is 910. Find the number of terms.

2.

The sum of the first n terms of the A.P. 20, 14, 8, ….. is -946, find the value of n.

3.

How many terms of the A.P. 12, 16, 20, ….. must be taken for the sum to be equal to 132.

4.

The sum of the first n term of an A.P. is given by Sn= n2 + 3n. Find (a) the first term, (b) the common difference, (c) the 10th term, (d) the nth term.

5.

The (a) (b) (c)

sum of the first n term of an A.P. is given by Sn= 5n - 3n2. Find the first term, the common difference, the second term, th (d) the n term.

6.

The (a) (b) (c)

sum of the first n term of an A.P. is given by Sn= n2 - 4n. Find the first term, the common difference, the 8th term, th (d) the n term.

7.

The second term of an A.P. is -1 and the sum of first five terms is -25. Calculate; (a) the first term, (b) the common difference, (c) the sum of the first eight terms.

8.

The third term of an A.P. is 11 and the sum of first six terms is 75. Calculate (a) the first term, (b) the common difference, (c) the sum of the first ten terms.

9.

The sum of the first five terms of an A.P. is 130 and the sum of the next five terms is 305. Find (a) the first term, (b) the common difference. (c) the sum of the first 20 terms.

10. An A.P. has 14 terms. The sum of the 14 terms is 301 while the sum of the odd term (1st, 3rd, 5th, .. etc) is 140. Find (a) the first term, (b) the common difference, (c) the last term.

11. A circle is divided into 8 sectors such that the sequence of the angle subtended at the centre of the circle for each sector forms an A.P. Given that the angle of the smallest sector is 10o, find (a) the common difference, (b) the angle of the largest sector, (c) the sum of the angles of the first six sectors, (d) the area of the fourth sector if the area of the whole circle is 36 cm2.

12.2 GEOMETRIC PROGRESSION (G.P.)
IMPORTANT POINTS 1. A geometric progression, G.P. is a number sequence where each tem after the first term is obtained by multiplying the preceding term by a constant. The constant is known as the common ratio. The common ratio, r, is defined by r=

2. 3.

T n 1 Tn

where Tn+1 = (n +1)th term, Tn = nth term.

4.

Steps to determine whether a given number sequence is an G.P.; (a) Take any three consecutive terms in a given number sequence; Tn, T n+1, T n+2. T T (b) Calculate the values of n 1 and n  2 Tn T n 1 (c) If

T n 1 Tn  2 = Tn T n 1

, then

the number sequence is a G.P..

(d) If

T n 1 Tn  2  then the number sequence is not a G.P. Tn T n 1

5.

The nth term, Tn of a G.P.;

Tn = ar n - 1

where a = first term, r = common ratio.

6. 7.

A G.P. can be written as a, ar, ar2 , ar3 , …… If the three consecutive terms of a G.P. are, a, b and c, b c then; =  b2 = ac a b

 b = ac , b is the geometric mean of a and c

8.

The sum of the first n terms, Sn, of a G.P.;

Sn 

a (r n  1) , for r > 1 r 1

or

Sn 

a (1  r n ) , for r < 1 1 r

where a = first term, r = common ratio.

9.

Sum to Infinity, S, of a G.P. For the case of -1 < r < 1, the sum of the first n terms of a G.P.;
Sn  a( 1  r n ) 1 r

,

when n  , r n   

S





a 1 r

,1<r<1

Exercise 12.3 1. Determine whether each of the following number sequences is a G.P. If so, state its common ratio. (c) 9x, 18x2, 36x 3, 72x 4, …..

(a) 48, 24, 12, 6, …..

(b) 4, -12, 36, -108, …..

(d) log10x, log10x2, log10x4, …..

2.

Determine the nth term, Tn, for each of the following G.P. (c) 8, 24, 72, ….. T12

(a) 40, 20, 10, ….. T8

(b) 28, -14, 7,….. T10

(d) -12, -6, -3, ….. T9

3.

Determine the number of terms, n of each of the following G.P.

(a) 6, -18, 54, ….., -13122.

(c) 96, 48, 24, …..

3 . 32

(b) 17, 34, 68, ….., 4352.

(d) -10, -5, -2½ , ….., -

5 . 256

4.

Calculate the sum of all the terms of each of the following G..P. (c) 32. -16, 8, ….. till the 10th term.

(a) 4, -12, 36 ….. till the 7th term.

(b) 3, 15, 75, ….. till the 8 th term.

(d) 30, 24, 19.2, ….. till the 12 th term.

5.

Calculate the number of terms, n, and the sum of all the terms, Sn, of each of the following G.P. (b) 14, -28, 56, ….., -448.

(a) 4, 12, 36, ….., 2916

(c) 64, 16, 4, …..,

1 . 256

(d) 480, 240, 120, …., 7.5.

6.

Calculate the number of terms, n, of each of the following G.P. (c) 7, -14, 28, …..; Sn = -2387.

(a) 6, 9, 13½, …..; Sn = 124.

(b) 4, 12, 36, …..; Sn = 39364.

(d) 48, 16, 5⅓, …..; Sn = 71

235 . 243

7.

Calculate the sum to infinity of the following G.P. (c) 80, -20, 5, …..

(a) 36, 12, 4, …..

(b) 42, 21, 10.5, …..

(d) 54, 21.6, 8.64, …..

8.

Find the first term, a, or the common ratio, r, of the following G.P. based on the given sum to infinity, S  . (a) S  = 36, r =
2 5

(c) S  = 24, a = 30

(b) S  = 72, r =

1 4

(d) S  = 32.5, a = 22.5

(e) S  = 48, r = - ⅓

(f) S  = 18, a = 12

9.

Write the following recurring decimals as a single fraction in its lowest terms. (d) 0.459459459…..

(a) 0.181818…..

(b) 0.242424..…

(e) 0.218218…..

(c) 4.135135…..

(f) 1.484848…..

Exercise 12.4 – Problem Solving II
1 and the sum to infinity is 24. 4

1.

Given that the common ratio of a G..P. is Calculate, (a) the first term. (b) the fourth term, (c) the sum of the first 6 term.

2.

Given that 6k + 8, 16, 2k are the first three terms of a G.P., find, (a) the values of k, (b) the sum of the first n terms, using the positive value of k.

3.

The price of a land property is RM220,000. Its price increases by 5% each year. Calculate the minimum number of years needed for the price to be more than RM400,000 for the first time.

4.

A boy saves 5 sen on the first day, 10 sen on the second day, 20 sen on the third day, and so on such that the amount of money he saves on each day is twice that of the previous day. Calculate the minimum number of days needed for the total amount of money to be more than RM1,000 for the first time.

5.

In a G.P., the sum of the second term and the third term is 12 while the sum of the third term and the fourth term is 6. Find the common ratio.

6.

The length of the sides of a triangle is in a geometric progression. The length of the longest side is 36 cm. It perimeter of the triangle is 76 cm. Find the positive value of the common ratio.

7.

The second term and the third term of a G.P. are

m3 4

and

m5 16

respectively.

(a) Find the values that cannot be taken by m apart from 0. (b) If m = 6, find (i) the first term and the common ratio, (ii) the sum of all the terms from the second term to the fifth term.

Exercise 12.5: SPM Past Years Questions
1. The first three terms of an arithmetic progression are k – 3, k + 3, 2k + 2. Find, (a) the value of k, (b) the sum of the first 9 terms of the progression.

[3 marks] SPM2003/Paper 1

Answer: (a) …………………………… (b) ……………………………

2.

In a geometric progression, the first term is 64 and the fourth term is 27. Calculate (a) the common ration, (b) the sum to infinity of the geometric progression. [4 marks] SPM2003/Paper 1

Answer: (a) …………………………… (b) ……………………………

3.

Given a geometric progression y, 2,

4 y

, p, ….. express p in term of y. [2 marks] SPM2004/paper 1

Answer: ………………………………..

4.

Given an arithmetic progression -7, -3, 1, ….., state three consecutive terms in this progression which sum up to 75. [3 marks] SPM2004/Paper 1

Answer: ………………………………..

5.

The volume of water in a tank is 450 liters on the first day. Subsequently, 10 liters of water is added to the tank everyday. Calculate the volume, in liters, of water in the tank at the end of the 7 th day. [2 marks] SPM2004/Paper 1

Answer: ……………………………….. Express the recurring decimal 0.969696….. as a fraction in its simplest form. [4 marks] SPM2004/Paper 1

6.

Answer: ………………………………..

7.

The first three terms of a sequence are 2, x, 8. Find the positive value of x so that the sequence is (a) an arithmetic progression. (b) an geometric progression. [2 marks] SPM2005/Paper 1

Answer: (a) …………………………… (b) ……………………………

8.

The first three terms of an arithmetic progression are 5, 9, 13. Find (a) the common difference of the progression. (b) the sum of the first 20 terms after the 3rd term. [4 marks] SPM2005/Paper 1

Answer: (a) …………………………… (b) …………………………… The sum of the first n terms of the geometric progression 8, 24, 72,……... is 8 744. Find (a) the common ratio of the progression, (b) the value of n. [4 marks] SPM2005/Paper 1

9.

Answer: (a) …………………………… (b) ……………………………

6cm 10. The about diagram shows part of an arrangement of bricks of equal size. The number of bricks in the lowest row is 100. For each of the other rows, the number of bricks is 2 less than in the row below. The height of each brick is 6 cm. Ali builds a wall by arranging bricks in this way. The number of bricks in the highest row is 4. Calculate (a) the height, in cm, of the wall, [3 marks] (b) the total price of the bricks used if the proice of one brick is 40 sen. [3 marks] SPM2005/Paper 2

y cm

x cm 11. The above diagram shows the arrangement of the first three of an infinite series of similar triangles. The first triangle has a base of x cm and a height of y cm. The measurements of the base and height of each subsequent triangle are half of the measurements of its previous one. (a) Show that the areas of the triangles form a geometric progression and state the common ratio. [3 marks] (b) Given that x = 80 cm and y = 40 cm. (i) determine which triangle has an area of 6
1 cm2. 4

(ii) find the sum to infinity of the areas, in cm2, of the triangles. [5 marks] SPM2004/Paper 2

12. The 9th term of an arithmetic progression is 4 + 5p and the sum of the first four terms of the progression is 7p – 10, where p is a constant. Given that the common difference of the progression is 5, find the value of p. [3 marks] SPM2006/Paper 1

Answer: ………………………………..

13. The third term of a geometric progression is 16. The sum of the third term and the fourth terms is 8. Find (a) the first term and the common ratio of the progression, (b) the sum to infinity of the progression. [4 marks] SPM2006/Paper 1

Answer: (a) …………………………… (b) ……………………………

14. Two companies, Delta and Omega, start to sell cars at the time. (a) Delta sells k cars in the first month and its sales increases constantly by m cars every subsequent month. It sells 240 cars in the 8 th month and the total sales for the first 10 months are 1 900 cars. Find the value of k and of m. [5 marks] (c) Omega sells 80 cars in the first month and its sales increase constantly by 22 cars every subsequent month. If both companies sell the same number of cars in the nth month, find the value of n. [2 marks] SPM2006/Paper 2


				
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