What is a series?
You can see that numbers in each of the series given above increase or decrease by a certain amount from the previous number, for example, numbers decrease by 2 in the first series, increase by 5 in the second series and increase by 4 in the third series. These types of number series in which there is a certain relation between successive numbers are known as progressions.
Find
out the pattern in each of the given progressions:-
(1)
4, 10, 16, 22, ..............................................
Pattern
4
+ 6 = 10
10
+ 6 = 16
16
+ 6 = 22
22
+ 6 = 28
28
+ 6 = 34
Ans.
4, 10, 16, 22, 28, 34
(2)
0, 3, 6, 9, ....................................................
Pattern
0
+ 3 = 3
3
+ 3 = 6
6
+ 3 = 9
9
+ 3 = 12
12
+ 3 = 15
15
+ 3 = 18
Ans.
3, 6, 9, 12, 15, 18
(3)
-1, -3, -5, -7, ..............................................
Pattern
(-1)
+ (-2) = -3
(-3)
+ (-2) = -5
(-5)
+ (-2) = -7
(-7)
+ (-2) = -9
(-9)
+ (-2) = -11
Ans.
-1, -3, -5, -7, -9, -11
Arithmetic progressions
You saw that in the series given above except the first term, each term is formed by adding a certain number to the previous term. These types of series of numbers are called Arithmetic Progressions or A.P. and the certain number added to each term is called common difference of the arithmetic progression. Common difference can be positive, negative or zero.
Look at the number series given below:
8, 13, 18, 23,
..............................................................................
The first term of this series is 8,
the second term is 13,
the third term is 18 and
the fourth term is 23.
Here, we get the next term by adding 5 to the previous term.
Therefore, the common difference of this progression is 5.
Example-1.
Find
the first term, the fourth term and the common difference for the given
arithmetic
progression.
-7,
-11, -15, -19.....
Solution:
First
term = -7, Fourth term = -19
Common
difference = Second term - First term
=
-11 - (-7)
=
-11 + 7
=
-4
1.
Find out which of the following sequences are arithmetic progressions:-
(i)
9, 16, 23, 30, ...........................
Pattern
First
term = 9
Second
term = 16
Third
term = 23
Fourth
term = 30
Here,
we get the next term by adding 7 to the previous term.
Common
difference = 7
Hence,
Given sequence is arithmertic progressions.
(ii)
11, 15, 18, 20, ...........................
Pattern
First
term = 11
Second
term = 15
Third
term = 18
Fourth
term = 20
There
is no common difference
Hence,
Given sequence is not a arithmertic progressions.
(iii)
4, 13, 19, 28, ...........................
Pattern
First
term = 4
Second
term = 13
Third
term = 19
Fourth
term = 28
There
is no common difference
Hence,
Given sequence is not a arithmertic progressions.
(iv)
0, -3, -6, -9, ...........................
Pattern
First
term = 0
Second
term = -3
Third
term = -6
Fourth
term = -9
Here,
we get the next term by adding (-3) to the previous term.
Common
difference = (-3)
Hence,
Given sequence is arithmertic progressions.
(v)
2, 2, 2, 2, ...........................
Pattern
First
term = 2
Second
term = 2
Third
term = 2
Fourth
term = 2
Here,
we get the next term by adding (0) to the previous term.
Common
difference = (0)
Hence,
Given sequence is arithmertic progressions.
(vi)
9(1/7), 7/7, 9/7, 13/7…………….
Pattern
First
term = 9(1/7)
Second
term = 7/7
Third
term = 9/7
Fourth
term = 13/7
There
is no Common difference
Hence,
Given sequence is not a arithmertic progressions.
2.
Write the first term and the common difference for the given arithmetic
progressions:-
(i)
9, 12, 15, 18, ................................
Solution:
First
term = 9, Second term = 12,
Third
term = 15, Fourth term = 18
Common
difference = Second term - First term
=
12 - 9
=
3
(ii) 2, 8, 14, 20,
..................................
Solution:
First
term = 2, Second term = 8,
Third
term = 14, Fourth term = 20
Common
difference = Second term - First term
=
8 - 2
=
6
(iii)
3, -2, -7, -12, ................................
Solution:
First
term = 3, Second term = -2,
Third
term = -7, Fourth term = -12
Common
difference = Second term - First term
=
(-2) - 3
=
-5
(iv)
-5, 2, 9, 16, ...................................
Solution:
First
term = -5, Second term = 2,
Third
term = 9, Fourth term = 16
Common
difference = Second term - First term
=
(-5) - 2
=
-7
(v)
0.4, 0.9, 1.4, 1.9, ...........................
Solution:
First
term = 0.4, Second term = 0.9,
Third
term = 1.4, Fourth term = 1.9
Common
difference = Second term - First term
=
0.9 – 0.4
=
0.5
(vi)
5, 5, 5, 5, .......................................
Solution:
First
term = 5, Second term = 5,
Third
term = 5, Fourth term = 5
Common
difference = Second term - First term
=
5 – 5
=
0
(vii)
1/3, 5/3, 9/3, 13/3, ………………….
Solution:
First
term = 1/3, Second term = 5/3,
Third
term = 9/3, Fourth term = 13/3,
Common
difference = Second term - First term
=
(5/3) – (1/3)
=
4/3