Sequences - MathsTips.com

Definition (Sequence):

A set of numbers arranged in a definite order according to some definite rule (or rules) is called a sequence. Each number of the set is called a term of the sequence. A sequence is called finite or infinite according as the number of terms in it is finite or infinite. The different terms of a sequence are usually denoted by $a_{1}, a_{2}, a_{3}, ...$ or by $T_1, T_2, T_3, ...$ .The subscript should be a natural number denoting the position of the term in the sequence. The number occurring at the nth place of a sequence i.e. $T_n$ is called the general term of the sequence.

A finite sequence is described by $a_1, a_2, ... , a_n$ or by $T_1, T_2, T_3, ... T_n$ and an infinite sequence is described by $a_1, a_2, a_3, ... \: to \: \infty$ or $T_1, T_2, T_3, ... \: to \: \infty$ . If all the terms are real, we have a real sequence; if all terms are complex numbers, we have a complex sequence etc.

For example, let us consider the following sequence:

1, 3, 5, 7,…, 21.

In the above sequence each term is obtained by adding 2 to the previous term. Also, this is a finite sequence. To define a sequence, we need not always have an explicit formula for the nth term. For the above sequence we may write $T_n=2+T_(n-1)$

If the terms of a sequence can be described by an explicit formula, then the sequence is called a progression.

Note: The sequence 1, 1, 2, 3, 5, 8, 13,…., is also a progression. It is called Fibonacci sequence.

Definition (Series):

If the terms of a sequence are connected by plus signs we get a series. Thus, if $a_1, a_2, a_3, ...$ is a given sequence then the expression $a_1+ a_2+ a_3, ...$ is called the series associated with the given sequence. The series is finite or infinite according as the given sequence is finite or infinite.

If $T_n$ denotes the general term of a sequence, then $T_1+ T_2+ T_3+ ... + T_n$ is a series of n terms. In a series $T_1+ T_2+ ... + T_k+ ...$ , the sum of first n terms is denoted by $S_n$ . Thus,

$S_n= T_1+ T_2+ ... +T_n= \sum\limits_{k=1}^{n} T_k = \sum T_n$

If $S_n$ denotes the sum of $n$ terms of a sequence, then

$S_n - S_{n-1}= (T_1+ T_2+ ... + T_n) - (T_1+ T_2+ ... +T_{n-1}) = T_n$

Thus, $T_n= S_n - S_{n-1}$ .

Example 1: Find the next term of the sequence

(i) 2, 4, 6, 8

(ii) 2, 8, 32, 128

(iii) -1, -3, -5, -7

(iv) 1, 8, 27, 64

Solution:

(i) We see that each term is obtained by adding 2 to the previous term. Hence next term= 8+ 2= 10.

(ii) We see that each term is obtained by multiplying the previous term by 4. Hence next term $= 128 \times 4= 512$

(iii) We see that each term is obtained by subtracting 2 from the previous term. Hence next term= -7-2 = -9

(iv) We see that terms are cubes of natural numbers – $1^3, 2^3, 3^3, 4^3$ . Hence next term = $5^3 = 125$

Example 2: Find the 18^th term of the sequence defined by $T_n= \dfrac{n(n-2)}{n+3}$ .

Solution:

Here $T_n= \dfrac{n(n-2)}{n+3}$ , putting $n= 18$ , we get,

$T_18= \dfrac{18(18-2)}{18+3} =\dfrac{18 \times 16}{21} =\dfrac{6 \times 16}{7} =\dfrac{96}{7}$ .

Example 3: Find the first five terms of the sequence given by $latexz a_1= 2, a_2=3+a_1 $ and $a_n=2. a_{n-1}+5$ for $n > 2$ .

Solution:

Here $a_1= 2, a_2= 3+a_1 = 3+2 =5$

Given $a_n=2. a_{n-1}+5$ for $n > 2$ , putting $n$ =3, 4, 5 we get

$a_3= 2. a_2+5 =2.5 + 5 =15$

$a_4= 2. a_3+5 =2.15 + 5 =35$

$a_5= 2. a_4+5 =2.35 + 5 =75$

Hence the first five terms of the given sequence are 2, 5, 15, 35, 75.

Example 4: If for a sequence ${a_n}, S_n = 2(3^n-1)$ , find its first four terms.

Solution:

Given,

$S_n = 2(3^n-1) \\ \Rightarrow S_{n-1} = 2(3^{n-1}-1) \\ \therefore a_n= S_n - S_{n-1} = 2(3^n-1)-2(3^{n-1}-1) \\ = 2(3^n-3^{n-1})= 2.3^{n-1}(3-1) = 4. 3^{n-1}$

Putting $n = 1, 2, 3, 4$ we get

$a_1= 4.3^0= 4, a_2= 4.3^1 = 12, a_3= 4.3^2= 36$ and $a_4= 4.3^3= 108$

Hence the first four terms of the sequence are 4, 12, 36, 108.

Example 5: Write $\sum\limits{k=1}^n (k^2+1)$ in expanded form.

Solution:

Putting k = 1, 2,3, 4, … , n in $(k^2+1)$ , we get 2, 5, 10, 17, … , n^2+1

Hence $\sum\limits{k=1}^n (k^2+1) = 2+ 5+ 10+ 17+ ... +(n^2+1)$ .

Exercise:

1) Write the first five terms of the sequences defined by

(i) $T_n = n (n-1)$

(ii) $T_n = \dfrac{n^2+1}{2n-3}$

2) Find the 18^th and 25^th terms of the sequence defined by

$T_n= n (n+2)$ , if $n$ is even natural number

$= \dfrac{4n}{n^2+1}$ , if $n$ is odd natural number.

3) A sequence is defined by $T_n=n^2 (n-1) (n-2)$ . Show that first two terms of the sequence are zero and the rest of the terms are positive.

4) The Fibonacci sequence is defined by $a_1= a_2= 1, a_n= a_{n-1}+ a_{n-2}$ for $n > 2$ . Find $\dfrac{a_n+1}{a_n}$ for $n = 1, 2, 3, 4, 5$ .

5) First term of a sequence is 1 and the $(n+1)th$ term is obtained by adding $(n+1)$ to the $nth$ term for all natural numbers $n$ . Find the sixth term of the sequence.