CBSE Class 10 Mathematics chapter 5 - ARITHMETIC PROGRESSIONS - Introduction

CBSE Class 10 Mathematics chapter 5 - ARITHMETIC PROGRESSIONS -  Introduction

Sequences, Series and Progressions

• Sequences:

A sequence is an ordered list of numbers, which can be finite or infinite and follow a specific pattern or rule. Various number occurring in this sequence called terms or element.

For example: 1, 2, 3, 4, 5... forms an infinite sequence of natural numbers.

• Series :

A series is the sum of the terms within the corresponding sequence.

For example: the series of natural numbers: 1 + 2 + 3 + 4 + 5... Each individual number within a sequence or series is called as a term.

• Progressions:

A progression is a type of sequence where the general term can be expressed using a mathematical formula.

• In the case of a finite sequence, it is conventionally represented as a₁, a₂, a₃, ... a_n, where 1, 2, 3, ..., n denotes the position of each term.
• When referring to the series, it is expressed as a₁ + a₂ + a₃ + ... + a_n, which is the sum of the sequence's terms.
• In the case of an infinite sequence, it is generally denoted as a₁, a₂, a₃, ... with the corresponding infinite series represented as a₁ + a₂ + a₃ + ...

Arithmetic Progression:

Definition:

1. An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This common difference is represented by 'd'.
2. 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.
3. It is abbreviated as AP.

Common Difference (d):

The common difference, d represents how the terms change as you move from one to the next. It can be positive, negative or zero.

Finding the Common Difference (d):

Let, the first term of an AP is a₁, second term is a₂, . . ., n^th term is a_n and the common difference is d. Then the AP becomes:

a₁, a₂, a₃, . . ., a_n

So, a₂ – a₁ = a₃ – a₂ = . . . = a_n – a_{n – 1} = d

Or we can write as:

If a is first term and d is the common difference, then:

• The difference between the 2nd and 1st terms is a₂ – a₁ = (a + d) —a = d
• The difference between the 3rd and 2nd terms is a₃ – a₂ = (a + 2d) —(a + d) = d
• The difference between the 4th and 3rd terms is a₄ – a₃ = (a + 3d) —(a + 2d) = d

So, the common difference d is the same for any pair of consecutive terms in the sequence

• a, a + d, a + 2d, a + 3d, . . .

So we can say that we must subtract any two consecutive terms in the sequence to find d.

Common difference between n^th term and n+1 term:

Following formula calculates the common difference (d) between the n^th term (a_n ) and the (n+1)^th term a_(n+1) in the AP. It tells us how much we need to add (or subtract) from one term to get to the next term in the sequence.

d = a_n — a_n+1

Where,

d = the common difference

a_n = the n^th term

a_n+1 = Previous term

1. If common difference (d) is positive (+), The AP is increasing.
2. If common difference (d) is Zero (0), The AP is constant.
3. If common difference (d) is Negative (—), The AP is decreasing.

The common difference between any two consecutive terms:

d = a_n+1 — a_n

This formula calculates the common difference (d) between the (n+1)^th term n_(n+1) and the n^th term (a_n) in the AP. It's used when we want to find the common difference between any two consecutive terms, not necessarily the n^th and (n+1)^th terms.

Note: To find d in the AP, we should subtract the n^th term from the (n + 1)^th term even if the (n + 1)^th term is smaller.

Types of Arithmetic Progressions (AP)

1. Finite Arithmetic Progression: A finite AP is contains finite number of terms and it has a last term. Example: 3, 5, 7, 9, 11, 13, 15, 17, 19
2. Infinite Arithmetic Progression: An infinite AP is a type of sequence that does not have a finite number of terms. It does not have a last term. Example: 5, 10, 15, 20, 25, 30, 35, 40, 45, and so on, extending indefinitely...

Identifying an AP:

To identify an AP, check if the difference between consecutive terms is constant.

If the common difference (d) remains the same, it's an AP.