Arithmetic Progressions

Arithmetic Progressions

An Arithmetic Progression is a sequence of numbers in which we get each term by adding a particular number to the previous term, except the first term.

• Each number in the sequence is known as term.

• The fixed number i.e. the difference between each term with its preceding term is known as common difference. It can be positive, negative or zero. It is represented as ‘d’.

General form of Arithmetic Progression

a, a+d, a+2d, a+3d,…..

Where the first term is ‘a’ and the common difference is ‘d’.

Example

Given sequence is 2, 5, 8, 11, 14,…

Here, a = 2 and d = 3

d = 5 – 2 = 8 – 5 = 11 – 8 = 3

First term is a = 2

Second term is a + d = 2 + 3 = 5

Third term is a + 2d = 2 + 6 = 8 and so on.

Finite or Infinite Arithmetic Progressions

1. Finite Arithmetic Progression

If there are only a limited number of terms in the sequence then it is known as finite Arithmetic Progression.

229, 329, 429, 529, 629

2. Infinite Arithmetic Progression

If there are an infinite number of terms in the sequence then it is known as infinite Arithmetic Progression.

2, 4, 6, 8, 10, 12, 14, 16, 18…..…

The n^th term of an Arithmetic Progression

If a_n is the nth term,a₁ is the first term, n is the number of terms in the sequence and d is a common difference then the nth term of an Arithmetic Progression will be

Example

Find the 11th term of the AP: 24, 20, 16,…

Solution

Given a = 24, n = 11, d = 20 – 24 = – 4

a_n = a + (n - 1)d

a₁₁ = 24 + (11-1) – 4

= 24 + (10) – 4

=24 – 40

= -16

Arithmetic Series

The arithmetic series is the sum of all the terms of the arithmetic sequence.

The arithmetic series is in the form of

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

Sum of first n terms of an Arithmetic series

Sum of the first n terms of the sequence is calculated by

Example

If Radha save some money every month in her piggy bank, then how much money will be there in her piggy bank after 12 months, if the money is in the sequence of 100, 150, 200, 250, ….respectively?

Solution

Given sequence is-

100, 150, 200, 250, …

a = 100 (first term)

d = 50 (common difference)

n = 12 (as we have to calculate money of 12 months)

Now we will put the values in the formula

So the money collected in her bank in 12 months is Rs. 4500.

But when we have finite Arithmetic Progression or we know the last term of the sequence then the sum of all the given terms of the progression will be calculated by

Where l = a + (n – 1)d i.e. the last term of the finite Arithmetic Progression.

Remark: The sum of the infinite arithmetic sequence does not exist.

Example

Find the sum of the sequence 38, 36, 34, 32, 30.

Solution

Given