Sequence And Series - Part 2

[leoumesh]

Hello and a warm and lovely greetings to all steemians. It's me @leoumesh and today I am gonna be continuing with the rest of topics related to sequence and series. If you are unfamiliar with this one, please Click Here to visit previous article where I had made an introductory explanation on sequence and series, their types along with formulae.

So lets begin with the rest of our topic.

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I. Arithmetico-Geometric Series

The terms of an AS are:

a , a+d , a+2d , a+3d , .......

The terms of a GS are:

1 , r , r² , r³ , .....

A series of the type

a.1 + (a+d)r + (a+2d)r² + (a+3d)r³ + .....
whose each term is the product of the corresponding terms of an AS and a GS is known as the arithmetico-geometric series

The following example will illustrate what is arithmetico-geometric series:

The given series may be written as follows:

Clearly, the above series is the arithmetic-geometric series with common difference = 1 and common ratio =

General Term of Arithmetico-Geometric Series

In the sequence
a.1 + (a+d)r + (a+2d)r² + (a+3d)r³ + .....

The n^th term of arithemtico-geometric series is given by multiplying the corresponding general terms of the arithmetic sequence (AS) and the geometric sequence (GS) as,

Sum of n terms

The sum of n terms of arithmetico-geometric series is given by,

S_n =

II. Sum of the Natural Numbers

The numbers 1 , 2 , 3 , 4 , ..... are said to be the natural numbers. Now , we derive some of the formulae for the sum of the first n natural numbers, the sum of the squares of the first n natural numbers and the sum of the cubes of the first n natural numbers.

1. Sum of the first n natural numbers

The first n natural numbers are 1 , 2 , 3 , ....... , n

Let ,

The sum of first n terms of the arithmetic series is given by S_n = [2a+ (n -1 )d]. So,

S_n
or, S_n =

2. Sum of the first n even natural numbers

The first n even natural numbers are 2 , 4 , 6 , ...... , n terms

Let ,
or,
or,

3. Sum of the firstn odd natural numbers

The first n odd natural numbers are 1 , 3 , 5 , ...... , n terms
Let ,

The sum of first n terms of the arithmetic series is given by S_n = [2a+ (n -1 )d]. So,

or,

4. Sum of the squares of the first n natural numbers

The sum of the squares of the first n natural numbers is given by the formula,

5. Sum of the cubes of the first n natural numbers

The sum of the cubes of the first n natural numbers is given by the formula,

The sum of the cubes of the first n natural numbers is the square of the sum of the first n natural numbers.

III. Means

A finite sequence consisting of more than two terms has one or more terms in between the first and the last terms. These terms are called the arithmetic , geometric or harmonic means according as the sequences are arithmetic , geometric or harmonic. In other words, we have the following definitions :

I. Any term in between the first and last terms of an arithmetic sequence is called an arithmetic mean(AM).

II. Any term in between the first and last terms of a geometric sequence is called a geometric mean(GM).

III. Any term in between the first and last terms of a harmonic sequence is called a harmonic mean(HM).

Formula for Means

Formula 1

Given any two numbers a and b , the AM , GM , and HM between them are given by :

1. AM = A =

2. GM = G =

3. HM = H =

Proof

1. If 'A' is the single AM between a and b , then a , A , b form an AS. Now by the defintion of an AS,

A - a = b - A
or, 2A = a + b
or, A =
A =

1. If 'G' is the single GM between a and b , then a , G , b form a GS. Now by the definition of a GS,

or,

1. If 'H' be a single HM between a and b, then a , H , b are in HP.

i.e. are in AP

Hence,
or,
or,

Formula 2

Given any two numbers a and b , the n AM's between them are given by,

a + d , a + 2d , a + 3d , a + 4d , ........ , a + nd

where ,

Proof

Let m₁ , m₂ , m₃ , ........... , m_n be n AM's to be inserted between a and b. Then,

a , m₁ , m₂ , m₃ , ........... , m_n , b are in AP

The number of terms in the above AS is n+2 of which the first term is a and the last term b, the (n + 2)^th term of an AS.

If d is the common difference then

b = a + (n + 2 - 1)d
or, b - a = (n + 1)d

Now,

m₁ = t₂ = a + d
m₂ = t₃ = a + 2d
m₃ = t₄ = a + 3d
.... ..... ..... .....
.... ..... ..... .....
m_n = t_n+1 = a + nd

Formula 3

Given any two numbers a and b , the n GM's between them are given by *ar , ar² , ar³ , ..... , arⁿ, where

Proof

Let G₁ , G₂ , G₃ , ...... , G_n be n GM's between a and b then,

a , G₁ , G₂ , G₃ , ...... , G_n , b form a GS

The number of terms in the above GS is (n+2) of which the first term is a and the last term b , the (n+2)^th term of a GS.

If r be the common ratio then ,

b = ar^{n+ 2 - 1}
or, b = ar^{n + 1}
or,

Now,

G₁ = t₂ = ar
G₂ = t₃ = ar²
G₃ = t₄ = ar³
.... ..... ..... .....
.... ..... ..... .....
G_n = t_{n + 1} = arⁿ

Formual 4

The AM , GM , HM between any two unequal positive numbers satisfy the following relations :

1. ( GM )² = ( AM ) * ( HM )

2. AM > GM > HM

Proof

Let a and b be two unequal positive numbers. Then,

AM =
( only positive square root is taken )
and,

To prove the first part, we have

AM * HM = *

or,
or,

AM * HM = ( GM )²

This result shows that GM is again the geometric mean between AM and HM.

To prove the second part , consider

AM - GM =

or, AM - GM =

or, AM - GM =

Which is always positive as we have squared the term.

Hence, AM > GM

Again,

( AM ) * ( HM ) = ( GM ) * ( GM )
or,
Since , AM > GM
We have, GM > HM

Combining the two , we have

AM > GM > HM

This result shows that AM , GM and HM are in decreasing order of magnitudes.

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Thank you for taking your time in reading this article. Please feel free to comment and interact.

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References

1. https://www.aplustopper.com/arithmetico-geometric-sequence/

2. https://www.careerbless.com/aptitude/qa/sequence_series_imp.php

3. Mathematical expression coded on : http://www.quicklatex.com/

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