Arithmetic Progression

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Introduction or Definition of Arithmetic Progression

An arithmetic progression occurs when the difference between every two consecutive terms in a series is the same. Or in simple terms, it is a sequence of numbers in which each number differs from the preceding one by a constant quantity.

The two basic definitions used to define arithmetic progression are as follows:

Definition 1: A mathematical sequence, denoted by the abbreviation AP, in which the difference between two subsequent terms is always a constant.

Definition 2: A progression or arithmetic sequence is a set of numbers where, for every pair of consecutive terms, the second number is obtained by adding a predetermined number to the first.

Let say the series is $1,5,9,13,17,...$

In the above series, we can see that when we subtract any two consecutive numbers, we get a difference of 4. Therefore, this is considered an arithmetic sequence of common difference 4 .

Notation

Arithmetic progression has two parts: the first term and the difference. Let’s take a look at both of them.

1. First Term:

This part of AP notation indicates the first term of an arithmetic sequence.

An AP can be expressed as:

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

If we express an arithmetic progression like above then the first term of this arithmetic progression will be 'a'.

2. Difference:

As we discussed earlier, an arithmetic sequence will give a constant value of subtraction of any two consecutive numbers that value is known as the common difference of an arithmetic sequence.

An AP can be expressed as:

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

If we express an arithmetic progression like above, then the common difference of this arithmetic progression will be 'd'.

Let's understand the notations with an example:

Let the series be $1,5,9,13,17,...$

In the above series, we can see that when we subtract any two consecutive numbers, we get a difference of 4. Therefore, this is considered an arithmetic sequence of common difference 4, and as the first element of this sequence is 1, then the first term of this arithmetic sequence will be 1.

First Term = 1 Difference = 4

General Form

In the general form of an arithmetic progression, we try to express arithmetic sequence in a general way in terms of its notations. Let's suppose we have an arithmetic sequence $x_1,x_2,x_3,x_4 … x_n$.

In this arithmetic sequence, we consider the first term '$x_1$' to be ‘a’ and the common difference 'd', which can be found out by $x_2-x_1$ or $x_3-x_2$.

Nth Term

The nth term indicates the formula to find the nth term in an arithmetic sequence.

Intuition:

In order to find the nth term in any sequence, we need to find the common point of all the linking elements, which are the first term and common difference. Here we observe that we can get any number in the sequence by adding a common difference to the first term a specific number of times . This insight helps us generate the formula for the nth term in an arithmetic sequence.

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Derivation:

The derivation is the same as finding the terms in general form. In which we simulate the formula for some elements and then observe that we can receive the nth term by adding the common difference (n-1) times to the first term.

Formula:

The formula to find the nth term in an arithmetic sequence is: $an = a+(n-1)*d$

• a = first Term 
• d = Difference

Let the series be $1,5,9,13,17,...$

In the above series, we can see that when we subtract any two consecutive numbers, we get a difference of 4. Therefore , this is considered an arithmetic sequence of common difference 4 and as the first element of this sequence is 1, then the first term of this arithmetic sequence will be 1.

Difference = 4 First Term = 1 Therefore, the $6th$ term in the series will be: 1+(6-1)4, i.e., 21.

Types of AP

The nth term indicates the formula to find the nth term in an arithmetic sequence .

There are 2 types of arithmetic progression. Let’s discuss them one by one.

Finite Arithmetic Progression (Finite AP):

As we can clearly understand from the name, an arithmetic progression series which has a last term, i.e., it ends somewhere, is known as Finite Arithmetic Progression.

Example of finite arithmetic progression is: $2,5,8,11,14$.

Infinite Arithmetic Progression (Infinite AP):

As we can clearly understand from the name, an arithmetic progression series which does not have a last term, i.e., it never ends, is known as infinite arithmetic progression.

Example of infinite arithmetic progression is: $1,2,3,4,5,6,7,8,9, ...$

Sum of AP for Nth term

The sum of n terms in an arithmetic progression series is referred to as the sum of all terms of that series till the nth term.

Intuition:

The Intuition to find the sum of the arithmetic progression series for the nth term was to simply convert each element of the series into simpler form with the help of arithmetic progression notations and then simply add them and compress them in order to generate a formula.

Derivation:

Consider an AP consisting of “n” terms having the sequence $a, a + d, a + 2d, ..., a + (n - 1) × d$

Sum of first n terms:

$a + (a + d) + (a + 2d) + ... + [a + (n - 1) × d]$ --- (i)

Writing the terms in reverse order, so we have:

$S_n$= $[a + (n - 1) × d] + [a + (n - 2) × d] + [a + (n - 3) × d] + ... (a)$ --- (ii)

Adding both the equations term wise, we have:

$2S_n$=$[2a + (n - 1) × d]+[2a + (n - 1) × d] + [2a + (n - 1) × d] + ... + [2a + (n - 1) ×d] (n-terms)$

$2S_n = {n[2a + (n - 1) × d]}$

$S_n = {n[2a + (n - 1) × d]\over 2}$

Formula:

The formula to find the sum of arithmetic progression series for nth term is:

$S_n = {n[2a + (n - 1) × d]\over 2}$, where a is a first term and d is a common difference.

In some cases, if the last term is given, we can also use this formula to find the sum of the arithmetic progression series:

$S = {n(FirstTerm + LastTerm)\over 2}$

Let’s consider the summation of 15 natural numbers.

AP = $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15$

In this case: a = 1, d = 1, and n = 15

Now, by using the formula:

$S_n$ = $n[2a + (n - 1) × d]\over 2$

$S_{15}$ = $15[2.1+(15-1)×1]\over 2$ = $15[2+14]\over 2$ = $15 * 8$ = $120$

Therefore, the sum of 15 natural numbers is 120.

Sum for First N Natural Numbers

We know that the natural numbers are also in the form of an arithmetic progression, with the first term as 1 and the common difference as 1. Therefore, in order to get a formula for the sum of n natural numbers, we simply place these values in the above formula.

$S_n$ = $n[2a + (n - 1) × d]\over 2$

$S_n$ = $n[2+(n-1)*1]\over 2$

$S_n$ = $n*[n+1]\over 2$

$S_n$ = $n*(n+1)\over 2$

To find the sum of the first 10 natural numbers: Sum = ${10*(10+1)\over 2}$ = ${10*11 \over 22}$ = 55