5.1 Introduction

You must have observed that in nature, many things follow a certain pattern, such as the petals of a sunflower, the holes of a honeycomb, the grains on a maize cob, the spirals on a pineapple and on a pine cone, etc.

We now look for some patterns which occur in our day-to-day life. Some such examples are :

(i) Reena applied for a job and got selected. She has been offered a job with a starting monthly salary of ₹ 8000 , with an annual increment of ₹ 500 in her salary. Her salary (in ₹) for the 1 st, 2nd, 3rd, . . . years will be, respectively

$ 8000, \quad 8500, \quad 9000, \ldots $

(ii) The lengths of the rungs of a ladder decrease uniformly by $2 cm$ from bottom to top (see Fig. 5.1). The bottom rung is $45 cm$ in length. The lengths (in $cm$ ) of the 1st, 2nd, 3 rd, …, 8th rung from the bottom to the top are, respectively

$ 45,43,41,39,37,35,33,31 $

Fig. 5.1

(iii) In a savings scheme, the amount becomes $\frac{5}{4}$ times of itself after every 3 years. The maturity amount (in ₹) of an investment of ₹ 8000 after 3, 6, 9 and 12 years will be, respectively :

$10000, \quad 12500, \quad 15625,19531.25$ (iv) The number of unit squares in squares with side $1,2,3, \ldots$ units (see Fig. 5.2) are, respectively

$ 1^{2}, 2^{2}, 3^{2}, \ldots $

Fig. 5.2

(v) Shakila puts ₹ 100 into her daughter’s money box when she was one year old and increased the amount by ₹ 50 every year. The amounts of money (in ₹) in the box on the 1st, 2nd, 3rd, 4th, . . . birthday were

100, 150, 200, 250, .., respectively.

(vi) A pair of rabbits are too young to produce in their first month. In the second, and every subsequent month, they produce a new pair. Each new pair of rabbits produce a new pair in their second month and in every subsequent month (see Fig. 5.3). Assuming no rabbit dies, the number of pairs of rabbits at the start of the 1st, 2nd, 3rd, …, 6th month, respectively are :

$ 1,1,2,3,5,8 $

Fig. 5.3

In the examples above, we observe some patterns. In some, we find that the succeeding terms are obtained by adding a fixed number, in other by multiplying with a fixed number, in another we find that they are squares of consecutive numbers, and so on.

In this chapter, we shall discuss one of these patterns in which succeeding terms are obtained by adding a fixed number to the preceding terms. We shall also see how to find their $n$th terms and the sum of $n$ consecutive terms, and use this knowledge in solving some daily life problems.

5.2 Arithmetic Progressions

Consider the following lists of numbers :

(i) $1,2,3,4, \ldots$

(ii) $100,70,40,10, \ldots$

(iii) $-3,-2,-1,0, \ldots$

(iv) $3,3,3,3, \ldots$

(v) $-1.0,-1.5,-2.0,-2.5, \ldots$

Each of the numbers in the list is called a term.

Given a term, can you write the next term in each of the lists above? If so, how will you write it? Perhaps by following a pattern or rule. Let us observe and write the rule.

In (i), each term is 1 more than the term preceding it.

In (ii), each term is 30 less than the term preceding it.

In (iii), each term is obtained by adding 1 to the term preceding it.

In (iv), all the terms in the list are 3 , i.e., each term is obtained by adding (or subtracting) 0 to the term preceding it.

In (v), each term is obtained by adding -0.5 to (i.e., subtracting 0.5 from) the term preceding it.

In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form an Arithmetic Progression ( AP ).

So, an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

This fixed number is called the common difference of the AP. Remember that it can be positive, negative or zero.

Let us denote the first term of an AP by $a_1$, second term by $a_2, \ldots, n$th term by $a_n$ and the common difference by $d$. Then the AP becomes $a_1, a_2, a_3, \ldots, a_n$. So, $\quad a_2-a_1=a_3-a_2=\ldots=a_n-a _{n-1}=d$.

Some more examples of AP are:

(a) The heights ( in $cm$ ) of some students of a school standing in a queue in the morning assembly are $147,148,149, \ldots, 157$.

(b) The minimum temperatures ( in degree celsius) recorded for a week in the month of January in a city, arranged in ascending order are

$ -3.1,-3.0,-2.9,-2.8,-2.7,-2.6,-2.5 $

(c) The balance money ( in ₹ ) after paying $5 %$ of the total loan of ₹ 1000 every month is $950,900,850,800, \ldots, 50$.

(d) The cash prizes ( in ₹ ) given by a school to the toppers of Classes I to XII are, respectively, 200, 250, 300, 350, …, 750.

(e) The total savings (in ₹) after every month for 10 months when ₹ 50 are saved each month are 50, 100, 150, 200, 250, 300, 350, 400, 450, 500.

It is left as an exercise for you to explain why each of the lists above is an AP.

You can see that

$ a, a+d, a+2 d, a+3 d, \ldots $

represents an arithmetic progression where $a$ is the first term and $d$ the common difference. This is called the general form of an AP.

Note that in examples (a) to (e) above, there are only a finite number of terms. Such an AP is called a finite AP. Also note that each of these Arithmetic Progressions (APs) has a last term. The APs in examples (i) to (v) in this section, are not finite APs and so they are called infinite Arithmetic Progressions. Such APs do not have a last term.

Now, to know about an AP, what is the minimum information that you need? Is it enough to know the first term? Or, is it enough to know only the common difference? You will find that you will need to know both - the first term $a$ and the common difference $d$.

For instance if the first term $a$ is 6 and the common difference $d$ is 3 , then the AP is

$ 6,9,12,15, \ldots $

and if $a$ is 6 and $d$ is -3 , then the AP is

$ 6,3,0,-3, \ldots $

Similarly, when

$ \begin{array}{lll} a=-7, & d=-2, & \quad \text{ the AP is }-7,-9,-11,-13, \ldots \\ a=1.0, & d=0.1, & \quad \text{ the AP is }-7,-9,-11,-13, \ldots \\ a=0, & d=1 \frac{1}{2},& \quad \text{ the AP is } 0,1 \frac{1}{2}, 3,4 \frac{1}{2}, 6, \ldots \\ a=2, & d=0,& \quad \text{ the AP is } 2,2,2,2, \ldots \end{array} $

So, if you know what $a$ and $d$ are, you can list the AP. What about the other way round? That is, if you are given a list of numbers can you say that it is an AP and then find $a$ and $d$ ? Since $a$ is the first term, it can easily be written. We know that in an AP, every succeeding term is obtained by adding $d$ to the preceding term. So, $d$ found by subtracting any term from its succeeding term, i.e., the term which immediately follows it should be same for an AP.

For example, for the list of numbers :

$ 6,9,12,15, \ldots, \\ $

We have

$ \begin{aligned} & a_2-a_1=9-6=3, \\ & a_3-a_2=12-9=3, \\ & a_4-a_3=15-12=3 \end{aligned} $

Here the difference of any two consecutive terms in each case is 3 . So, the given list is an AP whose first term $a$ is 6 and common difference $d$ is 3 .

For the list of numbers : $6,3,0,-3, \ldots$,

$ \begin{aligned} & a_2-a_1=3-6=-3 \\ & a_3-a_2=0-3=-3 \\ & a_4-a_3=-3-0=-3 \end{aligned} $

Similarly this is also an AP whose first term is 6 and the common difference is -3 .

In general, for an $AP a_1, a_2, \ldots, a_n$, we have

$ d=a _{k+1}-a_k $

where $a _{k+1}$ and $a_k$ are the $(k+1)$ th and the $k$ th terms respectively.

To obtain $d$ in a given AP, we need not find all of $a_2-a_1, a_3-a_2, a_4-a_3, \ldots$. It is enough to find only one of them.

Consider the list of numbers 1,1, 2, 3, 5, … By looking at it, you can tell that the difference between any two consecutive terms is not the same. So, this is not an AP.

Note that to find $d$ in the AP : $6,3,0,-3, \ldots$, we have subtracted 6 from 3 and not 3 from 6, i.e., we should subtract the $k$ th term from the $(k+1)$ th term even if the $(k+1)$ th term is smaller.

5.3 nth Term of an AP

Let us consider the situation again, given in Section 5.1 in which Reena applied for a job and got selected. She has been offered the job with a starting monthly salary of ₹ 8000 , with an annual increment of ₹ 500 . What would be her monthly salary for the fifth year?

To answer this, let us first see what her monthly salary for the second year would be.

It would be ₹ $(8000+500)=₹ 8500$. In the same way, we can find the monthly salary for the 3rd, 4th and 5th year by adding ₹ 500 to the salary of the previous year. So, the salary for the 3rd year $=₹(8500+500)$

$ \begin{aligned} & =₹(8000+500+500) \\ & =₹(8000+2 \times 500) \\ & =₹[8000+(\mathbf{3}-\mathbf{1}) \times 500] \quad \text{(for the 3rd year)} \\ & =₹ 9000 \end{aligned} $

Salary for the 4 th year $=₹(9000+500)$

$=₹(8000+500+500+500)$

$=₹(8000+3 \times 500)$

$=₹[8000+(4-1) \times 500] \quad$ (for the 4th year)

$=₹ 9500$

Salary for the 5 th year $=₹(9500+500)$

$ \begin{aligned} & =₹(8000+500+500+500+500) \\ & =₹(8000+4 \times 500) \\ & =₹[8000+(5-1) \times 500] \quad \text{ (for the 5th year) } \\ & =₹ 10000 \end{aligned} $

Observe that we are getting a list of numbers

$ 8000,8500,9000,9500,10000, \ldots $

These numbers are in AP. (Why?)

Now, looking at the pattern formed above, can you find her monthly salary for the 6th year? The 15th year? And, assuming that she will still be working in the job, what about the monthly salary for the 25th year? You would calculate this by adding ₹ 500 each time to the salary of the previous year to give the answer. Can we make this process shorter? Let us see. You may have already got some idea from the way we have obtained the salaries above.

Salary for the 15th year

$ \begin{aligned} & =\text{ Salary for the } 14 \text{ th year }+ \text{ ₹ } 500 \\ & =₹[8000+\underbrace{500+500+500+\ldots+500} _{13 \text{ times }}]+₹ 500 \\ & =₹[8000+14 \times 500] \\ & =₹[8000+(\mathbf{1 5}-\mathbf{1}) \times 500]=₹ 15000 \end{aligned} $

i.e., First salary $+(15-1) \times$ Annual increment.

In the same way, her monthly salary for the 25th year would be

$ \begin{aligned} & ₹[8000+(25-1) \times 500]=₹ 20000 \\ = & \text{ First salary }+(25-\mathbf{1}) \times \text{ Annual increment } \end{aligned} $

This example would have given you some idea about how to write the 15th term, or the 25th term, and more generally, the $n$th term of the AP.

Let $a_1, a_2, a_3, \ldots$ be an AP whose first term $a_1$ is $a$ and the common difference is $d$.

Then,

the second term $a_2=a+d=a+(2-1) d$

the third term $\quad a_3=a_2+d=(a+d)+d=a+2 d=a+(3-1) d$

the fourth term $\quad a_4=a_3+d=(a+2 d)+d=a+3 d=a+(\mathbf{4 - 1}) d$

Looking at the pattern, we can say that the $\boldsymbol{{}n}$ th term $a_n=a+(n-1) d$.

So, the $n$th term $a_n$ of the AP with first term $a$ and common difference $d$ is given by $a_n=a+(n-1) d$. $\boldsymbol{{}a} _{\boldsymbol{{}n}}$ is also called the general term of the AP. If there are $m$ terms in the AP, then $a_m$ represents the last term which is sometimes also denoted by $l$.

5.4 Sum of First $\boldsymbol{{}n}$ Terms of an AP

Let us consider the situation again given in Section 5.1 in which Shakila put ₹ 100 into her daughter’s money box when she was one year old, ₹ 150 on her second birthday, ₹ 200 on her third birthday and will continue in the same way. How much money will be collected in the money box by the time her daughter is 21 years old?

Here, the amount of money (in ₹) put in the money box on her first, second, third, fourth . . . birthday were respectively 100, 150, 200, 250, . . till her 21st birthday. To find the total amount in the money box on her 21st birthday, we will have to write each of the 21 numbers in the list above and then add them up. Don’t you think it would be a tedious and time consuming process? Can we make the process shorter? This would be possible if we can find a method for getting this sum. Let us see.

We consider the problem given to Gauss (about whom you read in Chapter 1), to solve when he was just 10 years old. He was asked to find the sum of the positive integers from 1 to 100 . He immediately replied that the sum is 5050 . Can you guess how did he do? He wrote :

$ S=1+2+3+\ldots+99+100 $

And then, reversed the numbers to write

$ S=100+99+\ldots+3+2+1 $

Adding these two, he got

$ \begin{aligned} 2 S & =(100+1)+(99+2)+\ldots+(3+98)+(2+99)+(1+100) \\ & =101+101+\ldots+101+101 \quad(100 \text{ times }) \end{aligned} $

So,

$ S=\frac{100 \times 101}{2}=5050 \text{, i.e., the sum }=5050 \text{. } $

We will now use the same technique to find the sum of the first $n$ terms of an AP :

$ a, a+d, a+2 d, \ldots $

The $n$th term of this AP is $a+(n-1) d$. Let $S$ denote the sum of the first $n$ terms of the AP. We have

$$ S=a+(a+d)+(a+2 d)+\ldots+[a+(n-1) d] \tag{1} $$

Rewriting the terms in reverse order, we have

$$ S=[a+(n-1) d]+[a+(n-2) d]+\ldots+(a+d)+a \tag{2} $$

On adding (1) and (2), term-wise. we get

$ 2 S=\underbrace{[2 a+(n-1) d]+[2 a+(n-1) d]+\ldots+[2 a+(n-1) d]+[2 a+(n-1) d]} _{n \text{ times }} $

or, $\quad 2 S=n[2 a+(n-1) d] \quad$ (Since, there are $n$ terms)

or, $\quad S=\frac{n}{2}[2 a+(n-1) d]$

So, the sum of the first $\boldsymbol{{}n}$ terms of an AP is given by

$ S=\frac{n}{2}[2 a+(n-1) d] $

We can also write this as

$ S=\frac{n}{2}[a+a+(n-1) d] $

i.e.,

$$ S=\frac{n}{2}(a+a_n) \tag{3} $$

Now, if there are only $n$ terms in an AP, then $a_n=l$, the last term.

From (3), we see that

$$ S=\frac{n}{2}(a+l) \tag{4} $$

This form of the result is useful when the first and the last terms of an AP are given and the common difference is not given.

Now we return to the question that was posed to us in the beginning. The amount of money (in Rs) in the money box of Shakila’s daughter on 1st, 2nd, 3rd, 4th birthday, …, were 100, 150, 200, 250, . . , respectively.

This is an AP. We have to find the total money collected on her 21st birthday, i.e., the sum of the first 21 terms of this AP.

Here, $a=100, d=50$ and $n=21$. Using the formula :

$ S=\frac{n}{2}[2 a+(n-1) d] $

we have

$ \begin{aligned} S & =\frac{21}{2}[2 \times 100+(21-1) \times 50]=\frac{21}{2}[200+1000] \\ & =\frac{21}{2} \times 1200=12600 \end{aligned} $

So, the amount of money collected on her 21st birthday is ₹ 12600 .

Hasn’t the use of the formula made it much easier to solve the problem?

We also use $S_n$ in place of $S$ to denote the sum of first $n$ terms of the AP. We write $S _{20}$ to denote the sum of the first 20 terms of an AP. The formula for the sum of the first $n$ terms involves four quantities $S, a, d$ and $n$. If we know any three of them, we can find the fourth.

Remark : The $n$th term of an AP is the difference of the sum to first $n$ terms and the sum to first ( $n-1$ ) terms of it, i.e., $a_n=S_n-S _{n-1}$.

5.5 Summary

In this chapter, you have studied the following points :

1. An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number $d$ to the preceding term, except the first term. The fixed number $d$ is called the common difference.

The general form of an AP is $a, a+d, a+2 d, a+3 d, \ldots$

2. A given list of numbers $a_1, a_2, a_3, \ldots$ is an AP, if the differences $a_2-a_1, a_3-a_2$, $a_4-a_3, \ldots$, give the same value, i.e., if $a _{k+1}-a_k$ is the same for different values of $k$.

3. In an AP with first term $a$ and common difference $d$, the $n$th term (or the general term) is given by $a_n=a+(n-1) d$.

4. The sum of the first $n$ terms of an AP is given by :

$ S=\frac{n}{2}[2 a+(n-1) d] $

5. If $l$ is the last term of the finite AP, say the $n$th term, then the sum of all terms of the AP is given by :

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

A Note TO THE READER

If $a, b, c$ are in AP, then $b=\frac{a+c}{2}$ and $b$ is called the arithmetic mean of $a$ and $c$.