Arithmetic Sequence

Introduction

 In our daily life, we often come across many things which follow a definite pattern or sequence.  Arrangement of leaves on the stem of a tree or the arrangement of grains on a cob of maize and the pattern of individual cells on a honeycomb are a few examples of patterns in nature. 

Arithmetic Progression is one such pattern. Over the coming few sections, we will learn about this beautiful concept in detail. 

What is an Arithmetic Sequence? 

An arithmetic sequence or arithmetic progression is a series of numbers, in which each term (number) is obtained by adding a fixed number to its preceding term. Though we never realize it, there are many instances of arithmetic sequences that we come across daily. Thinking of it, Mathematics itself is based on the arithmetic sequence. Let us see how!

• When we are counting numbers, we are following the Arithmetic Progression in which every successor can be obtained by adding one to its predecessor. 

\(\begin{align}1, 1+1 =2, 2 +1 =3, 3+1 = 4, 4+1 = 5….\end{align}\)

• Multiplication Tables are a brilliant example of Arithmetic Progression. We know that multiplication is a form of repeated addition. We have however seldom realized that it is also a form of AP in which the succeeding number is formed by adding a fixed value to its predecessor. 

The multiplication table of \(2\) can be formed by adding \(2\) repeatedly. Thus, 

\(\begin{align}&2 \times 1 = 2\\&2 \times 2 = 4\\&2 \times 3 = 6\\&2 \times 4 = 8\\&2 \times 5 = 10\\&....\\&.....\end{align}\)

• Factors and multiples also follow the sequence in an AP. All the multiples of a number can be obtained by adding the number to its preceding multiple. 

The multiples of \(3\) are

\(\begin{align}3, 6, 9, 12, 15, 18, 21, 24…\end{align}\)

For a given multiple, the next one can be obtained by just adding \(3\) to it. 

Applications of Arithmetic Progression in real life:

If you observe closely, you will notice how life around you moves closely in arithmetic sequences. Have a look at the following examples to check for yourself.

Time 

Have you ever observed the hands of a clock? The second's hand moves in Arithmetic Sequence, so does the minute's hand and the hour hand. Coming to think of it, even the weeks in a calendar follow the AP, so do the years. Each leap year can be known by adding \(4\) to the previous leap year. 

Celebration of people's birthdays 

Mathematics is so intricately weaved in our lives that we seldom realize it. The number of candles you blow on your birthday increases with arithmetic sequence every year!

The seats in a theatre 

Yes, they too are arranged in Arithmetic sequence! Next time when you are out for a movie, just observe how the seats in the theatre are arranged.

Stacking cups 

 When you are stacking the cups or building a house of cards, you are actually doing that in an arithmetic sequence. 

First-term of AP, Common Difference

Consider the following list of numbers:

1. \(\begin{align}50,100,150,200,….\end{align}\)
2. \(\begin{align}100,70,40,10,...\end{align}\)
3. \(\begin{align}3,3,3,3,...\end{align}\)
4. \(\begin{align}-3,-2,-1,0,...\end{align}\)

In all the above examples, each number except the first is obtained by adding a fixed number to its preceding number. Each number in the list or the series is called a term and the first number in the series is called the first term. The fixed number used to obtain each succeeding term is called the common difference. It is important to remember that the common difference can be positive, negative, or zero. 

Let us denote the terms of AP as \(\begin{align}{a_1},{a_2},{a_3},......\end{align}\) an, with \(\begin{align}{a_1}\end{align}\) as the first term and d is the common difference. 

Thus, \(\begin{align}{a_2} - {a_1} = {a_3} - {a_2} = \,...\,{a_n} - {a_n} - 1 = d.\end{align}\)

Thus, d can be obtained from the difference of two consecutive terms in an AP. 

In general, AP can be represented as:

\(\begin{align}a, a+d, a +2d, a+3d,...\end{align}\)

This is called the general form of an Arithmetic sequence. 

For the AP :  \(\begin{align}-6, -3, 0, 3, …\end{align}\) write the first term and the common difference. 

Solution:

 Arithmetic Sequence Formula

There are many ways to represent the arithmetic sequence formula. Let us study each of them:

Recursive formula 

The recursive formula explains how each term is related to its preceding one. As we know that the subsequent term is obtained by adding the common difference to its preceding term, we can represent the recursive formula as:

nth term = previous term + common difference. 

Generalizing this, we can represent it as

\(\begin{align}{a_n} = {a_{(n + 1)}} + d.\end{align}\)

Explicit formula 

The explicit formula helps us describe the arithmetic series formula such that the value of any term can be obtained. If the first term and common difference are known, we can very easily obtain any other term by repeated addition. 

Thus, nth term = first term + common difference × (number of terms from the first term). 
Generalizing this, we can write, 

\(\begin{align}{a_n} = {a_1} + d(n - 1)\end{align}\)

nth term formula

Let \(\begin{align}{a_1},{a_2},{a_3},....\end{align}\) be an AP with the first term as \(\begin{align}{a_1}\end{align}\) and common difference d. How to find the nth term? 

We can represent:

First term \(a\) as :- \(\begin{align}{a_1} = a + 0 + a + (1 - 1)d\end{align}\)

Second term \(\begin{align}{a_2}\end{align}\) as :- \(\begin{align}{a_2} = a + d = a + (2 - 1)d\end{align}\)

Third term \(\begin{align}{a_3}\end{align}\) as \(\begin{align}{a_3} = {a_2} + d = (a + d) + d = a + 2d = a + (3 - 1)d\end{align}\)

Fourth term \(\begin{align}{a_4}\end{align}\) as \(\begin{align}{a_4} = {a_3} + d = (a + 2b) + d = a + 3d = a + (4 - 1)d\end{align}\)
…….

…….

Extrapolating from the above pattern, we can now express the nth term \(\begin{align}{a_n}\end{align}\) as,

 \(\begin{align}{a_n} = a + (n - 1)d.\end{align}\)

Let us look at a solved example to understand the nth term formula better. 

Find the \(11th\) term of the AP: \(\begin{align}2,5,8,...\end{align}\)

Solution :

Given, \(\begin{align}a = 2, d = 5 - 2 = 3,\end{align}\) and \(n = 11\)

From the \(nth\) term formula, \(\begin{align}{a_n} = a + (n - 1)d\end{align}\)

Sum of Arithmetic Sequence

It is sometimes useful to know the arithmetic sequence sum formula for the first n terms. We can obtain that by the following two methods. 

When the values of the first term and the last term are known - In this case, the sum of arithmetic sequence or sum of an arithmetic progression is, 

\(\begin{align}{s_n} = \frac{n}{2}(a + {a_n})\end{align}\)

When the value of the first term and the common difference is given - In such case the sum of an AP can be obtained by the formula, 

\(\begin{align}{s_n} = \frac{n}{2}[2a + (n - 1)d]\end{align}\)

 where \(\begin{align}{s_n}\end{align}\) is the sum of \(n\) terms, 

\(n\)  is the number of terms, 

\(a\) is the first term, 

\(\begin{align}{a_n}\end{align}\) is the last term, and

\(d\) is the common difference. 

The sum of infinite arithmetic series is either \( \begin{align}+ \infty\end{align} \), if \(d>0\),  or \( \begin{align}- \infty\end{align} \), if \(d< 0\).

AP graph and Properties 

Let us have a look at a few common properties of AP 

• If a, b and c are three terms which are in AP, then

            \(2b = a +c\)

• If each term of an AP is increased, decreased, or multiplied with a constant number, then the series so formed will also be in an arithmetic series. 
• If the common difference is positive, then it results in an increasing arithmetic series
• If the common difference is negative, then it results in a decreasing arithmetic series. 

Graphically, the points \(\begin{align}n,{a_n}\end{align}\) of an AP can be plotted in the cartesian coordinate system. The points being collinear, result in a straight line graph. 

The slope of the graph is obtained as the common difference d. 

Summary

• An AP is a list of numbers in which each term is obtained by adding a fixed number to the preceding number. 
• a is represented as the first term, d is the common difference, \(\begin{align}{a_n}\end{align}\) as the nth term, and n as the number of terms. 
• In general,  AP can be represented as \(\begin{align}a, a + d, a + 2d, a + 3d,...\end{align}\)
• the nth term of an AP can be obtained as \(\begin{align}{a_n} = a + (n - 1)d\end{align}\)
• The sum of an AP can be obtained as either \(\begin{align}s_n = \frac{n}{2}(a + an)\end{align}\) or \(\begin{align}{s_n} = \frac{n}{2}[2a + (n - 1)d]\end{align}\)
• The graph of an AP is a straight line with the slope as the common difference. 

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