ARITHMETIC PROGRESSION (A. P)

ARITHMETIC PROGRESSION (A. P)

CONTENT

• Sequence
• Definition of Arithmetic Progression
• Denotations in Arithmetic progression
• Deriving formulae for the term of A. P.
• Sum of an arithmetic series

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

(a)        1, 5, 9, 13, 17, 21, 25(any term +4 = next term)

(b)        2, 6, 18, 54, 162, 486, 1458 (any term x 3 = next term)

(c)        1, 9, 25, 49, 81, 121, 169, (sequence of consecutive odd no)

(d)        10, 9, 7, 4, 0, -5, -11, –18, -26, (starting from 10, subtract 1, 2, 3 from immediate 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: 5^th term    = 4(5) -3

20 – 3 = 17

10^th term = 4(10) – 3

40 – 3 = 37

Example 2

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

(a)        (2n + 1)

(b)        3 – 5n.

Solution

(a)        2n + 1

            6^th term            =   2(6) + 1   = 12 + 1 = 13

9^th term            =   2 (9) + 1  = 18 + 1 = 19

(b)        3 – 5n

6^th term            = 3 – 5 (6) = 3 – 30 = -27

9^th term            = 3 – 5 (9) = 3 – 45 = –42

Evaluation

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

1 8

1 4

1 2

1. 1, , ,      ,       , ____,  ____

 

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 15^th 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          =          1^st term

d          =          common difference

n          =          no of terms

U_n         =          nth term

S_n         =          Sum of the first n terms

 

Formula for nth term of Arithmetic Progression

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

a          =          9

d          =          12 – 9   or   18 – 15 = 3.

1^st term            =          U₁         =          9                                  =    a

2^nd term           =          U₂         =          9 + 3                            =    a + d

3^rd term            =          U₃         =          9 + 3 + 3                     =    a + 2d

10^th term          =          U₁₀ = 9 + 9(3)                                      =    a + 9d

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

\nth term      =          U_n  = a + (n-1)d

 

Example:

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

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

U_n=  a + (n-1) d

U₅₀  =  9 + (50-1) 3

= 9 + (49) 3

= 9 + 147

= 156

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

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

U_n = 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   =?

U_n         =          a + (n-1) d

U₅₀       =          5 + (50-1)¹/₂

=          5 + 49(¹/₂)

U₅₀       =          5 + 24¹/₂

U₅₀       =          29¹/₂

Evaluation

1. Find the 37^th term of the sequence 20, 10, 0, -10…
2. 1, 5… 69 are the 1^st, 2^nd, and last term of the sequence; find the common difference between them and the number of terms in the sequence.

 

SUM OF AN ARITHMETIC SERIES

When the terms of a sequence are added, the resulting expression is called 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 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 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 sum of A. P. The following represent 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    U_n

2

or

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

2

Example 2

Find the sum of the 20^th 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

 

EVALUATION

1. Find the sum of the arithmetic series with 16 and -117 as the first and 20^th term respectively.
2. The salary scale for a clerical officer starts at N55, 200 per annum. A rise of N3, 600 is given at the end of each year; find the total amount of money earned in 12 years.

 

GENERAL EVALUATION /REVISION QUESTION

1. An A. P. has 15 terms and a common difference of -3, find its first and last term if its sum is 120.
2. On the 1^st of January, a student puts N10 in a box, on the 2^nd she puts N20 in the box, on the 3^rd she puts N30 and so on putting on the same no. of N10 notes as the day of the month. How much will be in the box if she keeps doing this till 16^th January?
3. The salary scale for a clerical officer starts at N55, 200 per annum. A rise of N3, 600 is given at the end of each year, find the total amount of money earned in 12 years.
4. 4. Find the 7^th  term  and  the  nth  term  of  the  progression  27,9,3,…
5. If 8, x, y, – 4 are in A.P, find x and y.

 

WEEKEND ASSIGNMENT

1. Find the 4^th 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 8^th and 4^th term is 20 and the 8^th term is 1¹/₂ times the 4^th 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

 

THEORY

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

2          a.         Write down the 15^th 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.
3. Find the 10^th  term    and  the  sum   of  the  first.

 

See also

PERCENTAGE ERROR

LOGARITHM OF NUMBERS

MENSURATION

Symmetric Properties of Roots

Sum & Product of Roots of a Quadratic Equation