Sequence And Series – Arithmetic Progression SS1 Mathematics Lesson Note

ARITHMETIC PROGRESSION

Find the next two terms in each of the following sets of numbers and in each case state the rule which gives the term.

1. 1, 5, 9, 13, 17, 21, 25 (any term +4 = next term)
2. 2, 6, 18, 54, 162, 486, 1458 (any term x 3 = next term)
3. 1, 9, 25, 49, 81, 121, 169, (sequence of consecutive odd no)

In each of the examples below, there is a rule which will give more terms in the list. A list like this is called a SEQUENCE in many cases; it can simply matter if a general term can be found for a sequence e.g.

1, 5, 9, 13, 17 can be expressed as

1, 5, 9, 13, 17 ……………. 4n – 3 where n = no of terms

Check: 5th term    = 4(5) -3

  20 – 3 = 17

10th term = 4(10) – 3

    40 – 3 = 37

Example 2

Find the 6th and 9th terms of the sequence whose nth term is

(a) (2n + 1)

(b) 3 – 5n.

Solution

(a) 2n + 1

6th term =   2(6) + 1   = 12 + 1 = 13

9th term =   2 (9) + 1  = 18 + 1 = 19

(b) 3 – 5n

6th term = 3 – 5 (6) = 3 – 30 = -27

9th term = 3 – 5 (9) = 3 – 45 = -42

CLASSWORK 

For each of the following sequences, find the next two terms and the rules which give the term.

1. 1,   ,½ ,  ¼    ,  ⅛     , ____,  ____

2 100, 96, 92, 88, _____, ____

3. 2, 4, 6, 8, 10,   ____, _____
4. 1, 4, 9, 16, 25,   ____, _____ 

(i) Arrange the numbers in ascending order   (ii) Find the next two terms in the sequence

5. 19, 13, 16, 22, 10
6. -21/2, 51/2, 31/2, 11/2, -1/2
7. Find the 15th term of the sequence whose nth term is 3n – 5

DEFINITION OF ARITHMETIC PROGRESSION

A sequence in which the terms either increase or decrease in equal steps is called an Arithmetic Progression.

The sequence 9, 12, 15, 18, 21, ____,  _____,  _____ has a first term of 9 and a common difference of +3 between the terms.

Denotations in A. P.

a = 1st term

d = common difference

n = no of terms

Un = nth term

Sn = The sum of the first n terms

The formula for the nth term of Arithmetic Progression

e.g. in the sequence 9, 12, 15, 18, 21.

a = 9

d = 12 – 9   or   18 – 15 = 3.

1st term = U1 = 9 =   a

2nd term = U2 = 9 + 3 =    a + d

3rd term = U3 = 9 + 3 + 3 =    a + 2d

10th term = U10 = 9 + 9(3) =  a + 9d

nth term = Un = 9+(n-1)3 = a + (n-1)d

(nth term =Un  = a + (n-1)d

Example:

1. Given the A.P, 9, 12, 15, 18 …… find the 50th term.

a   =  9        d    =  3      n    =  50         Un  =  U50

Un=  a + (n-1) d

U50  =  9 + (50-1) 3

        = 9 + (49) 3

        = 9 + 147

        = 156

2. The 43rd term of an AP is 26, find the 1st term of the progression given that its common difference is ½ and also find the 50th term.

U43=  26       d =  ½                a =  ?          n   = 43

Un = a + (n-1) d

26 = a + (43-1) ½ 

26 = a + 42(1/2)

26 = a + 21

26 – 21 = a

5 = a

a = 5

(b) a      =  5              d      = ½           n = 50     U50   =?

Un = a + (n-1) d

U50 = 5 + (50-1)1/2

= 5 + 49(1/2)

U50 = 5 + 241/2

U50 = 291/2

SUM OF AN ARITHMETIC SERIES

When the terms of a sequence are added, the resulting expression is called a series e.g. in the sequence 1, 3, 5, 7, 9, 11.

Series = 1 + 3 + 5 + 7 + 9 + 11

When the terms of a sequence are unending, the series is called an infinite series, it is often impossible to find the sum of the terms in an infinite series.

e.g.  1 + 3 + 5 + 7 + 9 + 11 + …………………. Infinite

Sequence with the last term or nth term is termed finite series.

e.g. Find the sum of

1, 3, 5, 7, 9, 11, 13, 15

If sum = 2, n = 8

Then 

S = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 

Or S = 15 + 13 + 11 + 9 + 7 + 5 + 3 +1

Add eqn1 and eqn 2

2s = 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16

=  48   =    8(16)

                2              2        =    S   = 64

DERIVING THE FORMULA FOR THE SUM OF A. P. 

The following represents a general arithmetic series when the terms are added.

S = a + (a+d) + a + 2d + …………………………… + (L-2d) + (L-d) + L – eqn

S = L + (L-d) + L – 2d + ……………………………… a + 2d + (a+d) + a – eqn

2s = (a + L) + (a + L) + (a + L) + …………………… (a + L) + (a + L) + (a + L)

2s = n(a + L)

             2

S = n(a+L)

          2    

L   =>  Un  =  a + (n-1)d

Substitute L into eq**

S = n(a + a+(n-1)d

                 2   

S = n(2a + (n-1)d    = n ( 2a+ (n-1)d

22

(  S = n[a + L]    where   L  is the last  term i.e    Un

2 I’ll

or

S  =n[2a +(n-1)d]  when d is given or obtained

Example 2

Find the sum of the 20th term of the series 16 + 9 + 2 + …………………

a = 16           

d = 9 – 16 = -7           

n = 20

S = n(2a + (n-1)d)

               2

S = 20 (2×16) + (20-1)(-7)

2  

=     20 (32 + 19(-7)

                2

S =10 (32 – 133) = 10(-101)

S = -1010

ASSIGNMENT

1. Find the 4th term of an A. P. whose first term is 2 and the common difference is 0.5   

(a) 4   (b) 4.5    (c) 3.5     (d) 2.5

2. In an A. P. the difference between the 8th and 4th term is 20 and the 8th term is 11/2 times the 4th term, find the common difference         

(a) 5    (b) 7     (c) 3    (d) 10

3. Find the first term of the sequence in no. 2        

(a) 70     (b) 45    (c) 25     (d) 5

4. The next term of the sequence 18, 12, 60 is       

(a) 12     (b) 6    (c) -6    (d) -12

5. Find the no. of terms of the sequence 1/2, ¾, 1, ……………….. 51/2     

(a) 21    (b) 43/4      (c) 1     (d) 22

6. Eight wooden poles are to be used for pillars and the length of the poles forms an arc Arithmetic Progression (A. P.) If the second pole is 2m and the 6th pole is 5m, give the lengths of the poles in order and sum up the lengths of the poles.

7. i. Write down the 15th term of the sequence.

  2_,  3   ,4  ,   5 

1×3   2×4    3×5   4 x6

1. An arithmetic progression (A. P.) has 3 as its term and 4 as the common difference.
2. Write an expression in its simplest form for the nth term.

d.Find the 10th term and the sum of the first