ICSE Class 10 Maths Chapter 10 Arithmetic Progression

10 Arithmetic Progression

Introduction

A group of numbers, which are arranged in a definite order following a certain rule, is called a sequence.

For example:

1. 2, 4, 6, 8, ........... is a sequence in which each number is 2 more than its preceding number.

2. 2, 4, 16, 256, ........... is a sequence in which each number is square of its preceding number.

3. 30, 27, 24, ........... is a sequence in which each number (term) is 3 less than its preceding term.

4. 5, 8, 6, 15, ........... is not a sequence as numbers of this group do not have any definite relation with their preceding numbers.

The numbers used in a sequence are called its elements or terms.

When the numbers (terms) in a sequence are connected to each other by positive (plus) sign or negative (minus) sign, the sequence becomes series.

Thus, (i) 2 + 4 + 6 + 8 + ..................... is a series.

(ii) - 2 - 4 - 6 - 8 ..................... is a series.

(iii) 2 + 4 + 16 + 256 + ..................... is a series and so on.

Sequence and series are used in the same sense.

When the members (elements) of a series (or, sequence) are written in a definite order, following certain rule, we get a progression.

For example:

3, 7, 11, 15 ..................... is a progression and successive term of 15 is 15 + 4 = 19, successive term of 19 is 19 + 4 = 23 and so on.

Similarly, 5 + 15 + 45 + 135 + ..................... is progression.

In a progression, its terms (elements) are in such a pattern which can suggest the successor of every element of it.

10.2 Arithmetic Progression (A.P.)

An arithmetic progression is a sequence (series) of numbers in which each term can be obtained by adding a certain quantity to its preceding term.

For example, the sequence 3, 8, 13, 18, 23, ........... is an arithmetic progression in which every term (other than the first term) can be obtained by adding 5 to its preceding term i.e.

3 + 5 = 8, 8 + 5 = 13, 13 + 5 = 18 and so on.

In other words, we can say that the difference between any two consecutive terms of the sequence, given above, is 5.

In an A.P., the difference between two consecutive terms is called its common difference and is denoted by letter d.

Thus, if \(t_1, t_2, t_3, t_4, ...............\) are terms (numbers) in A.P., its first term is \(t_1\) and is denoted by a.

So, \(a = t_1\)

\(d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = ................\) and so on.

Each of the following series is an arithmetic progression.

1. 4, 6, 8, 10, ................

with first term a = 4 and common difference d = 6 - 4 = 2.

2. 4 + 8 + 12 + 16 + ................

with a = 4 and common difference d = 8 - 4 = 4.

3. -4 - 6 - 8 - 10 ................. has a = -4 and d = -6 - (-4) = -2.

4. 28, 25, 22, 19, ................. has a = 28 and d = 25 - 28 = -3.

Conversely, if first term (a) of an A.P. is 8 and its common difference (d) is 3, the A.P. is:

8, 8 + 3, 8 + 3 + 3, 8 + 3 + 3 + 3, .................

= 8, 11, 14, 17, ................

More examples:

Problem 1

Is the sequence 12, 8, 4, 0, .................. an A.P.?

If yes; state its first term and common difference.

Solution:

Since, 8 - 12 = -4, 4 - 8 = -4 and 0 - 4 = -4

- Difference between consecutive terms is the same.

- The given sequence is an A.P.

Clearly, first term = 12 and common difference = -4

Problem 2

For the A.P. 7, 15, 23, 31, .................., write the first term, common difference and next two terms.

Solution:

First term of the given A.P. = 7. Its common difference = 15 - 7 = 8

Next two terms are 31 + d = 31 + 8 = 39 and 39 + 8 = 47

Teacher's Note

Arithmetic progressions appear in everyday life - such as calculating monthly loan payments that decrease by a fixed amount, or plant growth that increases by a consistent amount each week.

10.3 General Term Of An Arithmetic Progression

Let the first term of an A.P. be 'a' and its common difference be 'd', the terms of the A.P. can be taken as:

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

or,

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

First term = a = a + (1 - 1)d = a + (no. of term - 1)d

Second term = a + d = a + (2 - 1)d = a + (no. of term - 1)d

Third term = a + 2d = a + (3 - 1)d = a + (no. of term - 1)d

On proceeding in the similar manner, we find:

10th term = a + (10 - 1)d = a + 9d

24th term = a + (24 - 1)d = a + 23d

5th term = a + 4d, 12th term = a + 11d and so on.

On combining all the discussions made in this article, we find:

nth term of an A.P. = a + (n - 1)d i.e. \(t_n = a + (n - 1)d\)

Here, \(t_n = a + (n - 1)d\) is called the general term of the A.P. in which by putting n = 1, 2, 3, etc., we can get first term, second term, third term, respectively of the A.P. under consideration.

If an A.P. has only n terms, its nth term (last term) is denoted by letter l.

Thus for an A.P. with first term = a, common difference = d and number of terms = n, we have \(l = a + (n - 1)d.\)

For an A.P., \(t_n = a + (n - 1)d\)

- if d > 0, the A.P. is increasing,

- if d < 0, the A.P. is decreasing and

- if d = 1, all the terms of the A.P. are the same.

Thus, for A.P. of n terms,

A.P. = a + (a + d) + (a + 2d) + ........................

= a + (a + d) + (a + 2d) + ........................ + [a + (n - 1)d]

= a + (a + d) + (a + 2d) + ........................ + l

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

Problem 3

Find the 20th term of the sequence: 9, 5, 1, -3, ........ .

Solution:

Since, 5 - 9 = -4, 1 - 5 = -4, -3 - 1 = -4, etc. the given sequence is an A.P. with first term a = 9 and common difference d = -4.

\(t_n = a + (n - 1)d \Rightarrow t_{20} = 9 + (20 - 1) \times -4\)

= 9 + 19 \times -4 = 9 - 76 = -67

Problem 4

Find the A.P. whose second term is 12 and 7th term exceeds the 4th by 15.

Solution:

Second term = 12 - a + d = 12

7th term - 4th term = 15 - (a + 6d) - (a + 3d) = 15

i.e. 3d = 15 - d = 5 and a + d = 12 - a = 7

- The required A.P. = a, a + d, a + 2d, a + 3d, ........................

= 7, 7 + 5, 7 + 2 × 5, 7 + 3 × 5, ....................

= 7, 12, 17, 22, ....................

Teacher's Note

Understanding the general term formula helps in predicting values in sequences - like determining what salary you'll earn in your 10th year if it increases by a fixed amount annually.

This is a preview of the first 3 pages. To get the complete book, click below.