Sequence and Series

Sequence

A Sequence is a list of things (usually numbers) that are in order. let us consider the following of numbers.

(i) 1, 4, 7, 10,.....

(ii) 20, 18, 16, 14,....

(iii) 1, 3, 9, 27, 81,....

(iv) 1, 2, 3, 4, ...

We observe that each term after the first term

(i) is formed by adding 3 to the preceding term;

(ii) is formed by subtracting 2 from the preceding term; 

(iii) is formed by multiplying the preceding term by 3; each term in

(iv) is formed by squaring the natural numbers 1, 2, 3, 4,.....

In all the above case, we see that set of number follow a certain rule and we can easily say what number will come next to given number. thus, the numbers come in succession in accordance with a certain rule or low. A succession of numbers formed and arranged in a definite order according to a certain definite rule is called a sequence. the successive number in a sequence are called its terms.

Series

A series is formed by adding or subtracting the successive term of a sequence. A series is finite or infinite according to as the number of terms added in the corresponding sequence is finite or infinite.

eg. 1 + 4 + 7 + 10 + .......... + 25 is a finite series and 2 + 4 + 6 + 8 + ........... is finite series.

The successive numbers forming the series are called the terms of the series and the successive terms are denoted by t₁, t₂, t₃,....., t_n ,which denotes the 1^st, 2^nd, 3^rd, ...... n^th term respectively. The n^th term, t_n, of a series, is called its general term. Thus, in a series, 1+4+7+10+.......+25, the first term is 1, the second term is 4, the third term is 7, and so on.

Progression

A sequence of number is said to be a progression if the difference or ratio between its two successive terms is constant throughout the whole sequence. An example of progression is as follows.

(i) 1, 3, 5, 7,..... (ii) 1, 3, 9, 27,.....

In (i), the difference between two successive terms is equal to 2.

In (ii), the ratio of two successive terms is equal to 3.

Types of Progression

Progression is divided into following two types.

(i) Arithmetic progression

(ii) Geometric progression

Arithmetic progression or Sequence

A sequence is called an arithmetic progression if the difference between its two successive terms is constant throughout the whole sequence. An arithmetic progression can be denoted by A.P. The constant number obtained by subtracting succeeding term from its preceding term is called the common difference.

For example:-

(i) 1, 3, 5, 7, 9,....

(ii) 15, 12, 9, 6,......

From (i), we find that

second term - first term = 3 -1 = 2,

third term - second term = 5 - 3 = 2,

fourth term - third term = 7 - 5 = 2 and so on.

From (ii), we find that

second term - first term = 12-15 = -3,

third term - second term = 9 - 12 = -3,

fourth term - third term = 6 - 9 = -3 and so on.

Hence, the common difference 'd' is calculated by 

d =  succeeding term - proceeding term = t_n - t_n-1

Here, we find that the difference between two successive terms, in both sequences, are same or constant. So, such sequence is called arithmetic progressions. The C.D. of the two progressions are 2 and -3 respectively. Thus, arithmetic progressions is a series in which the successive terms increase or decrease by the common difference.

General term or n^th term of an A.P.

To find the n^th term of an A.P.

Let, t₁ be the first term, n be the number of terms and 'd' the common difference of an A.P. respectively. Then,

t₁ = a = a + (1-1)d

t₂ = a + d = a + (2-1)d

t₃ = a + 2d = a + (3-1)d

t₄ = a + 3d = a + (4-1)d

In general, t_n = a + (n-1)d

Formula: If t_n denotes the n^th term, of the arithmetic progression whose first term, common term and number of terms are a, d and n respectively.

With this term, arithmetic sequence and series can be written as:

Arithmetic sequence: a, a+d, a+2d, a+3d, ............

Arithmetic series: a+ (a+d) + (a+2d) + (a+3d), ..........

Arithmetic Mean

The terms between the arithmetic progression are known as arithmetic mean. Such as the three numbers 2, 4, 6 are in arithmetic progression with the common difference d = 2, then 4 is the arithmetic mean between 2 and 6.

For example:

Let a, b,c are in arithmetic progression

 b-a = c-b

or, b+b = a+c

or, 2b = a+c

or, b = \(\frac{a+c}{2}\)

Hence the arithmetic mean between a and c is (\(\frac{a+c}{2}\))

n Arithmetic Means between two numbers a and b

Let m₁, m₂, m₃, .........m_n be the arithmetic means between the given term a and b. Then, a, m₁, m₂, m₃, .........m_n, b are in A.P.

Here, numbers of arithmetic means = n

So, numbers of terms of A.P. = n+2

It means, 

    b = (n+2)^th term of AP

or, b = a + (n+2-1)d, where d is common difference

or, b =a + (n+1)d

or, (n+1)d = b-a

∴ d= \(\frac{b-a}{n+1}\)

Now, m₁ = a+d = a +  \(\frac{b-a}{n+1}\)

m₂ = a + 2d = a +  \(\frac{2(b-a)}{n+1}\)

m₃ =  a +  \(\frac{3(b-a)}{n+1}\)

.............................................

m_n =  a +  \(\frac{n(b-a)}{n+1}\)

Sum of n terms of series in A.P.

Let us consider an arithmetic series

a + (a+d) + (a+2d) + (a+3d) + ...... + (l-2d) +(l-d) + l

Here, the first term = a, 

first term = a, 

common difference = d,

number of terms=  n,

last term (t_n) = l

the term before last term = l-d

if the sum of n terms is denoted by S_n, then

S_n = a + (a+d) + (a+2d) + (a+3d) + ...... + (l-2d) +(l-d) + l .... (i)

Writing term in the reverse order,

S_n = l + (l-d) + (l-2d)  + ...... + (a+3d) + (a+2d) + (a+d) + a .... (ii) 

Adding the corresponding terms of (i) and (ii)

\(\frac{S_n \;= \;a \;+\; (a+d)\; +\; (a+2d)\; +\; (a+3d)\; + \;......\; + \;(l-2d) \;+\; (l-d)\; + \;l\\S_n\;=\;l\; +\;(l-d)\;+\;(l-2d)\;+\;......\;+\;(a+3d)\;+\;(a+2d)\;+\;(a+d)\;+a\:}{2S_n\;= \;(a+l) \;+ \;(a+l)\; + \;(a+l)\; +\; ............ \;+\; (a+l)\; + \;(a+l)\; +\; (a+l)}\)

=   n times (a+l)

=   n (a+l)

= \(\frac{n}{2}\)(a+l)

But, the last term l = a + (n-1)d

So, S_n = \(\frac{n}{2}\)(a+l) = \(\frac{n}{2}\)[a+a+(n-1)d] = \(\frac{n}{2}\)[2a+(n-1)d]

∴ S_n =  \(\frac{n}{2}\)[2a+(n-1)d]

Thus, if d is unknown, S_n = \(\frac{n}{2}\)(a+l)

And, if l is unknown, S_n = \(\frac{n}{2}\)[2a+(n-1)d]

1. Sum of first n natural numbers

the numbers 1, 2, 3, 4, ......, n are called the first n natural numbers. 

Here, first term (a) = 1

Common difference (d) = 2-1 = 1

Number of terms (n) = n

If S_n denotes the sum of these first n natural numbers, then

S_n = \(\frac{n}{2}\)[2a+(n-1)d] = \(\frac{n}{2}\)[2.1+(n-1).1] =  \(\frac{n}{2}\)[2+n-1] = \(\frac{n}{2}\)(n+1)

2. Sum of first n odd numbers

1, 3, 5, 7, ......., (2n-1) are the first n odd numbers.

Here, first term (a) = 1

Common difference (d) = 3-1 = 2

Number of terms (n) = n

If S_n denotes the sum of these first n odd numbers, then

S_n = \(\frac{n}{2}\)[2a+(n-1)d] = \(\frac{n}{2}\)[2.1+(n-1).2] =  \(\frac{n}{2}\)(2+2n-2) =  \(\frac{n}{2}\) × 2n = n²

3. Sum of first n even numbers

2, 4, 6, 8, ......., 2n are the first n even numbers.

Here, first term (a) = 2

Common difference (d) = 4-2 = 2

Number of terms (n) = n

If S_n denotes the sum of these first n even numbers, then

S_n = \(\frac{n}{2}\)[2a+(n-1)d] = \(\frac{n}{2}\)[2.2+(n-1).2] =  \(\frac{n}{2}\)(4+2n-2) =  \(\frac{n}{2}\)(2n-2) = n(n+1)

Geometric Progression or Sequence

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly,10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

Examples of a geometric sequence are powers r^k of a fixed number r, such as 2^k and 3^k. The general form of a geometric sequence is

a, ar, ar², ar³, ............arⁿ

where r ≠ 0 is the common ratio and a is a scale factor equal to the sequence's start value.

General Term or n^th term of G.P.

We use the following notations for terms and expression involved in a geometrical progression:

The first term = a the n^th term = toor b

The number of terms = n Common ratio = r.

The expression ar^n-1 gives us the n^th term or the last term of the geometric progression whose first term, common ratio and a number of terms are a, r and n respectively.

∴t_n = ar^n-1

With the help of this general term, geometric sequence and series can be written in the following ways:

Geometric Sequence: a, ar, ar², ar\(^3\), .....

Geometric series: a + ar + ar² + ar\(^3\) + ........

Geometric Mean

If the three numbers are in G.P., then the middle term is called the geometric mean of the other two terms. In other words, the geometric mean of two non-zero numbers is defined as the square root of their product.

Let a, G, b be three numbers in G. P., then the common ratio is the same i.e.

\(\frac{G}{a}\) =\(\frac{b}{G}\)

or, G² = ab

or, G =\(\sqrt{a}{b}\)

Hence, the geometric mean of two numbers a and b is the square root of their product i.e. \(\sqrt{a}{b}\).

So, the geometric mean between two number 2 and 8 is G =\(\sqrt ab\) = \(\sqrt2*8\) = \(\sqrt16\) = 4.When

When any number of quantities are in G. P., all the terms in between the first and last terms are called the geometric means between these two quantities.

Here, G_n = arⁿ = a \(\begin{pmatrix}b\\a\\ \end{pmatrix}\)^{\(\frac{n}{n + 1}\)}

Relation between arithmetic mean and geometric mean

"Arithmetic mean (A. M) is always greater than Geometric mean (G. M.) between two position real unequal numbers".

Let us consider two numbers 2 and 8

Here, AM between 2 and 8 =\(\frac{2 + 8}{2}\) = 5

GM between 2 and 8 = \(\sqrt 2 * 8\)) = 4

∴ AM > GM.

The sum of n terms of a series in G. P.

Let us consider geometric series a + ar + ar² + ar\(^3\) + .......+ ar^{n -3}+ arⁿⁱ²+ ar^n-1

Here, first = a common ratio = r number of terms = n last term (l) = ar^n-1

∴ S_n = \(\frac{lr - a}{r - 1}\)

If the number of terms is odd, we take the middle term as aand the common ratio as r. If the number of terms is even, we take \(\frac{a}{r}\) and ar as the middle terms and r² as the common ratio.