have observed many things in nature that follow a certain pattern, such as the
holes of a honeycomb, the grains on a maize cob, the spirals on a pineapple
often observe interesting patterns of numbers in which quantity increases or
decreases progressively in our day-to-day life. Some such examples
Example 1: Ria
applied for a job and got selected. She has been offered a job with a starting
monthly salary of rupees 10,000 with an annual increment of rupees 1000 in her
salary. Her salary (in rupees) for the 1st, 2nd, 3rd …… years will
12,000, and so on
In a saving scheme the amount becomes 2 times of itself after every 3 years. The
maturity amount (in rupees) of an investment of rupees 10,000 after 3, 6, 9 and
12 years will be, respectively: 20,000, 40,000, 80,000,
and so on
the examples above, we observed that the succeeding terms are obtained by
adding a fixed number in the first example and by multiplying with a fixed
number in the second example. These patterns are generally known as sequences.
Therefore, a sequence
is an arrangement of numbers in a definite order according to some rule. Example:
4, 6, 8, 10… Here numbers are even natural number.
we will study a particular type of sequences which are known as arithmetic
arithmetic progression is a list of numbers in which each term is obtained by
adding a fixed number to the preceding term except the first term.
-3, -2, -1, 0 …..
of the numbers in the list is called a term.
We denote the terms of a sequence by a1,
a2, a3 ….etc. or x1,
x2, x3 …etc. Here, the subscripts denote the
positions of the terms. The first term of the A.P is denoted by a1 and the number at the
second place is called the second term and is denoted by a2 and so on. In general,
the number at the nth place is called the nth term of the sequence and
is denoted by an. The
term is called the general term of the sequence. In the above example a1
(first term) = -3, a2 (second term) = -2, a3 (third
term) = -1, a4 (fourth term) = 0, and so on.
Each term is obtained by adding a fixed
number except first term. Here in the above example each term is obtained by
adding 1 to the term preceding it. And we can write the other next term by
adding same fixed number. This fixed number is called the common difference of the A.P (Arithmetic Progression). It can be positive,
negative or zero. Remember that we can find d using any two consecutive terms, once
we know that the numbers are in AP. In the above example d = an –an-1 (difference
between the next term and previous term of the A.P.) i.e. a2 – a1 = -2 – (-3) =
Example 2: Let
us consider the sequence of even natural numbers 2, 4, 6, 8, 10 …..
(first term) = 2 = 2x1
a2 (second term) = 4 = 2x2
a3 (third term) = 6 = 2x3
a4 (fourth term) = 8 = 2x4
a5 (fifth term) = 10
and so on.
is evident from this that an (nth term) = 2xn =
n = 1, 2, 3…..)
Example 3: Let
us now consider the sequence of squares of natural numbers i.e., 1, 4, 9, 16,
(first term) = 1 = 12
a2 (second term) = 4
a3 (third term) = 9 = 32
a4 (fourth term) = 16
a5 (fifth term) = 25
and so on.
is evident from this that an (nth term) = n2
(where n = 1, 2, 3…..)
Let us study the sequence of odd natural numbers i.e., 1, 3, 5, 7, 9 ….
(first term) = 1 = 2x1-1
a2 (second term) = 3
a3 (third term) = 5 =
a4 (fourth term) = 7
a5 (fifth term) = 9 =
and so on.
is evident from this that an (nth term) = 2xn-1
= 2n-1 (where n = 1, 2, 3…..)
We can also describe a sequence by writing
the algebraic formula for its nth term or general term. In some cases,
the terms of the sequence do not follow some fixed pattern but they are
generated by some recursive relation.
Consider for instance, the sequence, 1, 2, 3, 5, 8 …..
(first term) = 1
a2 (second term) = 2
a3 (third term) = 2 =
1+2 = a1 + a2
a4 (fourth term) = 3
= 1+3 = a2 + a3
a5 (fifth term) = 5 =
2+3 = a3 +a4 and so on.
It is evident from this that an
(nth term) = an-1 + an-2 for n>2
For the AP: 3, 1, -1, -3…. write the first term a1 and the common
Here, first term (a1) = 3
Common difference (d) = an –an-1
Example 7: Is 4, 10, 16, 22 ….an
A.P? If they form an A.P, write the next two terms.
d = a2 – a1
= 10 – 4 = 6
d = a3 – a2 = 16 – 10 = 6
d = a4 – a3 = 22 – 16 = 6
We can see that all the value of common
difference is fixed or equal.
Hence, 4, 10, 16, 22….an A.P and next two
terms are 22 + 6 = 28 and 28 + 6 = 32.