Consider the following pictures
There is some pattern or arrangement in these pictures. In the first picture, the first row contains one apple, the second row contains two apples and in the third row there are three apples etc... The number of apples in each of the rows are 1, 2, 3, ...
In the second picture each step have 0.5 feet height. The total height of the steps from the base are 0.5 feet,1 feet, 1.5 feet,... In the third picture one square, 3 squares, 5 squares, ....
These numbers belong to category called “Sequences”.
A real valued sequence is a function defined on the set of natural numbers and taking real values
Each element in the sequence is called a term of the sequence. The element in the first position is called the first term of the sequence. The element in the second position is called second term of the sequence and so on’
If the nth term is denoted by an, then a1 is the first term, a2 is the second term, and so on.
A sequence can be written as a1, a2, a3, …… an, …
1. 1,3,5,7,... is a sequence with general term an = 2n − 1 . When we put n = 1, 2, 3,..., we get a1 =1, a2 = 3, a3 = 5, a4 = 7,...
2. 1/2 , 1/3 , 1/4 , 1/5 ,... is a sequence with general term 1/ [n + 1] . When we put n = 1,2,3,.... we get
a1 = 1/2 , a2 = 1/3 , a3 = 1/4 , a4 = 1/5 ,...
If the number of elements in a sequence is finite then it is called a Finite sequence. If the number of elements in a sequence is infinite then it is called an Infinite sequence
A sequence can be considered as a function defined on the set of natural numbers N. In particular, a sequence is a function f : N → R , where R is the set of all real numbers.
If the sequence is of the form a1,a2,a3,... then we can associate the function to the sequence a1,a2,a3,... by f (k) = ak , k = 1,2,3,...
Find the next three terms of the sequences
(i) 1/16 , 1/6 , 1/14 , . . . . (ii) 5, 2,- 1, -4,. . . . (iii) 1, 0.1, 0.01,. . .
In the above sequence the numerators are same and the denominator is increased by 4.
So the next three terms are
Here each term is decreased by 3. So the next three terms are -7, -10, -13 .
Here each term is divided by 10. Hence, the next three terms are
Though all the sequences are functions, not all the functions are sequences.
Find the general term for the following sequences
(i) 3, 6, 9,... (ii) 1/2 , 2/3 , 3/4 ,... (iii) 5, -25, 125,...
(i) 3, 6, 9,...
Here the terms are multiples of 3. So the general term is
a n = 3n,
(ii) 1/2 , 2/3 , 3/4 ,...
a1 = 1/2 ; a2 = 2/3 ; a3 = 3/4
We see that the numerator of nth term is n, and the denominator is one more than the numerator.
Hence, an = n / [n+ 1] , n ∈ N
(iii) 5, -25, 125,...
The terms of the sequence have + and – sign alternatively and also they are in powers of 5.
So the general term a n = (−1)n +1 5n ,n ∈ N
The general term of a sequence is defined as
Find the eleventh and eighteenth terms.
To find a11 , since 11 is odd, we put n = 11 in an = n(n + 3)
Thus, the eleventh term a11 = 11(11 + 3) = 154 .
To finda18 , since 18 is even, we put n = 18 in a n = n2 + 1
Thus, the eighteenth term a18 = 18 2 + 1 = 325.
Find the first five terms of the following sequence.
The first two terms of this sequence are given bya1 = 1 , a2 = 1. The third term a3 depends on the first and second terms.
Similarly the fourth term a4 depends upon a2 and a3 .
In the same way, the fifth term a5 can be calculated as
Therefore, the first five terms of the sequence are