ARITHMETIC PROGRESSION

a, a+d, a+2d, a+3d, a+4d, …

+ d + d + d + d

EXAMPLE

3, 7, 11, 15 …

ARITHMETIC PROGRESSION

n

n

The sum of the first n term:

S = ______________

or

S = ______________

Fill in the blank

a = _______________d = _______________l = _______________n = _______________

Find out the formula form your note.

1.1 Identify characteristics of arithmetic progression:

EXERCISE 1:

Complete each of the sequence below to form an arithmetic progression.

a) 2, _____ , 8 , ______ , 14 , 17

b) -2 , -5 , ______ , -11 , ________

c) -4, ______ , ______ , ________ , 2

d) 2x -3 , 2x -1 , ________ , _______ , 2x + 5

1

2

1.2 Determine whether a given sequence is an arithmetic progression

EXERCISE 2 :

1. Determine whether a given sequence below is an arithmetic progression.

a) 5 , 11 , 17 , 23 , ……….. ( ____________________ )b) -20 , -50 , -30 , -35 , ………. ( ____________________ ) c) 1 , 4 , 9 , 16 , ……… ( ____________________ )d) 2x + y , 4x – y , 6x -3y , …… ( ____________________ )

example

Identify whether the following is an arithmetic progression by calculating the common difference d.

4, 7, 10, 13, …= 7 – 4 = 31d

= 10 – 7 = 32d

= 13 – 10 = 33d

Common difference, d = 3.

4, 7, 10, 13, … is an arithmetic progression.

2. k +3 , 2k + 6 ,8 are the first three terms of an arithmetic progression,find the value of k.

Note:If x,y and z are three terms of an arithmetic progression , y – x = z - y

3.Given that are three consecutive terms of

an arithmetic progression where x has a positive value. Find the value of x.

2 ,5 ,7 4x x x

4.Given that the first three terms of

an arithmetic progression are

Find the value of y.

2 ,3 3 and 5y+1y y

example

Determine the specific 10-th term for the following AP.

-20, -14, -8, -2, …

( 1)nT a n d

1.3 Determine by using formula:

a. specific terms in arithmetic progressionsb. the number of terms in arithmetic progressions

9th

9 2 (9 1)3T

1. Find the term of the AP.

a = 2d = 5-2=3

= _______

2, 5 , 8 , ….. Solution:

( 1)nT a n d

Other method,

list out the AP,

2, 5, 8, 11, 14,

17, 20, 23, 26

11th

53, , 2,........

2

2. Find the term of the arithmetic prog.

3. For the arithmetic progression 0.7, 2.1 , 3.5, ….. , find the term. 5th

thn1

4,6 ,9,.....2

4. Find the term of the arithmetic progression

5. Find the 7th term of A.P.:

k, 2k + 1, 3k+2, 4k+3,….

6. Given that arithmetic progression 9 +6x , 9+4x, 9+2x, …… Find the 10th term.

7.Find the number of terms of the arithmetic progression

a) 4,9.14,….. ,64

64nT 4 + (n-1)5 = 64 4+ 5n-5 = 64 5n – 1 = 64 5n = 65 n = 65 ÷ 5 n = 13

( 1)nT a n d

Other method,

list out the AP,

4, 9, 14, 19,

24, 29, 34, 39,

44, 49, 54, 59,

64 n = 13

7.Find the number of terms of the arithmetic

progression (b) -2, -7, -12, ……, -127

( 1)nT a n d

7.Find the number of terms of the arithmetic

progression

( 1)nT a n d c) 1 1 1

1,1 ,1 ,......, 46 3 2

7.Find the number of terms of the arithmetic

progression

d) , , 3 ,........., 31x y x y x y x y

8. For the arithmetic progression 5 , 8 , 11 , ……. Which term is equal to 320 ?

a =

d =

Solution:

( 1)nT a n d nT=

5 + ( n – 1)(3) = 320

5 + 3n – 3 = 320

2 + 3n = 320

3n = 320 - 2

3n = 318

n = 106

The sequence 121 , 116 , 111 , …. is an arithmetic progression. Find the first negative term in the progression.

9.

Solution:

nT < 0

a + (n – 1)d < 0

a = 121,

d = 116 – 121 = -5

121 + (n – 1)(-5) < 0

121 - 5n + 5 < 0

126 - 5n < 0

126 < 5n

5n > 126

n > 25.2

n = 26

Other method,

list out the AP,

121, 116, 111, 106, 101, 96,

91, 86, 81, 76, 71, 66,

61, 56, 51, 46, 41, 36,

31, 26, 21, 16, 11, 6,

1, -4 n = 26

Find the number of terms of the arithmetic progression

2 ,3 3 ,5 4 ,......, 23 13a b a b a b a b

10.

Solution: nT

List all the terms,

a + 2b

3a + 3b

5a + 4b

7a + 5b

9a + 6b

11a + 7b

13a + 8b

15a + 9b

17a + 10b

19a + 11b

21a + 12b

23a + 13b

12th term

Answer: n = 12

11. Find the value of the first term a and the common difference d. Then, calculate the 8th term of the progressions.

36,15 52 TT(a)

a + d = 15

Solution:

a + 4d = 36

3d = 21

d = 7

from

a + d = 15

a + 7 = 15

a = 15 – 7

= 8

a + d = 15

= 8 + (8 – 1)(7)8TdnaTn )1(

= 8 + (7)(7)

= 57

11. Find the value of the first term a and the common difference d. Then, calculate the 8th term of the progressions.

(b) 112,138 53 TT

(c) nTn 25

(d) 12, 10215 TTTT

12. Calculate the number of terms in each arithmetic progressions where the last term of the progression is given. (a) 143, 134, 125, 116,…, 62 Solution:

( 1)nT a n d

62 = 143 + (n – 1)(-9)

a =143d = 134 – 143 = - 9

143 + (n – 1)(-9) = 62

143 - 9n + 9 = 62

152 - 9n = 62

- 9n = 62 - 152

- 9n = - 90

n = 10

Other method,

list out the AP,

143, 134, 125, 116, 107, 98,

89, 80, 71, 62

n = 10

12. Calculate the number of terms in each arithmetic progressions where the last term of the progression is given.

(b) 526...,,511,58,55,52

(c)

(d)

7, 3, -1, -5,…, -37

1.2, 2.4, 3.6, …, 24

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ?

1 + 2 + 3 + …… + 98 + 99 + 100 = ?

1 + 2 + 3 + …… + 48 + 49 + 50

+ 100 + 99 + 98 + …… + 53 + 52 + 51

50 groups of 101

101 101 101 101 101 101

1 + 2 + 3 + …… + 98 + 99 + 100 = 50 x 101

= 5050

ARITHMETIC PROGRESSION

We have more sistematic way to solve the sum, that calls

Find a) the sum of the first n terms of arithmetic progressions.

b) the sum of a specific number of consecutive terms of arithmetic progressions.

c) the value of n , given the sum of the first n terms of arithmetic progressions.

),(2

lan

Sn or

S = sum

n = number

a = first term

l = last term

d = common diference

Exerise 1.41. Find the sum of the first 12 terms of the arithmetic progression -10 , -7 , -4, ……

)3)(11()10(22

1212 S a = -10,

d = -7-(-10)

= 3= 78

)7)(8()38(22

99 S

a = 38

d = 31 - 38

= -7

2. Find the sum of all the terms of the 38 , 31 , 24 , …., -18

)(2

lan

Sn

18)7)(1(38 nTn

38 – 7n + 7 = -18

45 – 7n = -18

45 + 18 = 7n

7n = 63

n = 9

= 90

or )18(382

99 S

= 90

B u t

You should find value n first.

Now, you can find the sum.

Other methodList out all the terms.

38, 31, 24, 17, 10, 3, -4, -11, -18

9S =38+31+ 24+17+10

+3+(-4)+(-11)+(-18)

= 90

)( 54321 TTTTT

)5)(13()4(22

1414 S

)5)(4()4(22

55 S

a = -4

d = 5

= 30

Other methodList out all the terms.

-4, 1, 6, 11, 16, 21, 26, 31, 36, 41,46, 51, 56, 61

3. For the arithmetic progression -4 , 1 , 6 , …….find the sum of all the terms from the 6th term to the 14th term.

sum from the 6th term to the 14th term.

14131211109876 TTTTTTTTT

14S

14S5S

= 399

sum of all the terms from the 6th term to the 14th term

= 399 – 30

=369

146 TTS

= 21+26+31+36+…+56+61

= 369

How many terms of the arithmetic 3, 8 , 13 ,18 , …… must be taken for the sum to be equal to 1575?

SOLUTION:

a = 3 , d = 6

525n

15756662

nn

Sn

= 1575 )6)(1()3(22

nn

Sn

15753 n

157562

nn

Solve problems involving

arithmetic progressions.

SOLUTION:

1689 daT

dnaTn )1(

445 daT

1689 daT

3d

d4

The 5th and 9th of arithmetic progression are 4 and 16 respectively.Find the 15th term.

4)3(4 a

12

445 daT

Substitute d=3 into a + 4d = 4

a + 12 = 4 a = 4 – 12 a = -8

)3)(115(815 T

)3)(14(8

34

Other method

The 5th and 9th of arithmetic progression are 4 and 16 respectively.Find the 15th term.

4, ____, ____, ____, 16 w, x, y, z

102

164

10

72

104

7

d = 3

w = 7-3 -3-3-3-3

w = -8

)3)(115(815 T

)3)(14(8

34

The 2nd and 8th terms of an AP are -9 and 3 respectively. Fin the first term and the comman difference.And the sum of 10 terms beginning from the 12th term.

The sum of the first 6 terms of an arithmetic progression is 39 and the sum of the next 6 terms is -69. Find a) the first term and the common difference b) the sum of all the terms from the 15th term to the 25th term.

. The sum of the first n terms is given by

26 3nS n n

Finda)the nth term in terms of nb)the common difference

3, 3,2 2k k k

The first three terms of an arithmetic progression are

Finda) the value of kb) the sum of the first 9 terms of the progression

Given an arithmetic progression -7, -3, 1, …. , state three consecutive terms in this progression which sum up to 75

The volume of water in a tank is 450 litres on the first day. Subsequently, 10 litres of water is added to the tank everyday.Calculate the volume , in litres , of water in the tank at the end of the 7th day.