Class 11 Maths Sequences Series Arithmetic Progression

Arithmetic Progression

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, sequence 1, 3, 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2. Each of the numbers in the list is called a term.

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term. E.g.: 2,4,6,8,10 …. This fixed number is called the common difference of the AP. This common difference can be positive, negative or zero

  • Positive common difference (3): 1,4,7,10,13..
  • Negative common difference (-1) : 7,6,5,4,3,2,1,0,-1 …
  • Zero Common Difference : 3,3,3,3,3,3,3,3,3,3….

Let us denote the first term of an AP by a1, second term by a2, . . ., nth term by an and the common difference by d. Then the AP becomes a1, a2, a3, . . ., an.

So, a2 – a1 = a3 – a2 = . . . = an – an – 1 = d.

Thus a, a + d, a + 2d, a + 3d, . . . represents an arithmetic progression where a is the first term and d the common difference. This is called the general form of an AP.

 

AP with a finite number of terms are called a finite AP E.g.: {1,3,5,7}, while AP with infinite number of terms is called infinite AP, E.g.:{1,3,5,7,…}.

 

To know about an AP, the minimum information that we need is

  1. First term, denoted by ‘a’
  2. Common difference, denoted by ‘d’

With this we can form the AP as a, a + d, a + 2d, a + 3d, ……

For instance if the first term a is 5 and the common difference d is 2, then the AP is 5, 7,9, 11, . . .

Also, given a list, we can tell if it is AP or not.

List 2,4,6,8,10… here 4-2 = 6-4 = 8-6 = 10-8 =2, thus it is AP with common difference 2

List 1,3,4,6,7, here 3-1 ≠ 4-3, thus it is not AP.

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