Chapter 5: Arithmetic Progression CBSE Math Notes

Chapter 5: Arithmetic Progression (AP)

Sequence

➥ A sequence is an arrangement of numbers in a definite order according to some rule.
For example:

1. 1, 2, 3, 4, 5, 6, … (Sequence of natural numbers)
2. 3, 5, 7, 9, 11, …
3. -8, -5, -2, 1, 4, 7, …
4. 6, 1, -4, -9, 14, …
5. 3+√2, 3+2√2, 3+3√2, 3+4√2, …

Arithmetic Progression (A.P.)

➥ An arithmetic progression (A.P.) is a sequence of numbers in which each term is obtained by adding a fixed number (called the common difference) to the preceding term.

Example:
2, 4, 6, 8, 10, 12, … (Common difference = 2)

Common Difference

➥ The common difference (d) is the difference between two consecutive terms in an A.P. It can be positive, negative, or zero.

Example:
For the A.P. 2, 6, 10, 14, 18, 22, ... the common difference is 4.

How to Find Common Difference?

To find the common difference, subtract any term from its preceding term.

Example:
For the A.P. 4, 7, 10, 13, 16, ...:
Common difference, d = 7 - 4 = 3 or 10 - 7 = 3, and so on.

General Form of A.P.

The general form of an arithmetic progression is:

General form: a, (a + d), (a + 2d), (a + 3d), …
Where 'a' is the first term, and 'd' is the common difference.

Finite and Infinite A.P.

1. Finite A.P.: An A.P. with a fixed number of terms.
   Example: 2, 5, 8, … 35, 38
2. Infinite A.P.: An A.P. with infinite terms.
   Example: 2, 5, 8, 11, …

The nth Term in an A.P.

General Term of A.P.: The nth term of an A.P. is given by the formula:
Tn = a + (n - 1)d
Where 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.

Example: Finding the 48th Term, A.P.: 1, 4, 7, 10, 13, …

Given A.P.: 1, 4, 7, 10, 13, …
The 4th term is 10, but if you need to find a higher term, use the formula: Tn = a + (n - 1) d
For n = 48, a = 1, d = 3: T48 = 1 + (48 - 1)x3 = 1 + 141 = 142

Sum of n Terms of an A.P.

➯ The sum of the first 'n' terms of an A.P. is given by:

Sum of n terms:
Sn = n/2 [2a + (n - 1)d]
Example:
For the A.P. 1, 4, 7, 10, 13, ... with a = 1, d = 3, and n = 48:
Sn = 48/2 x [2 x 1 + (48 - 1) x 3]
= 24 x 143 = 3432

Sum of an A.P. is given, then-

➯ The sum of the first 'n' terms of an A.P. is given by:

Sn = n/2 [a + l]
where, a = first term
l = Last term
n= The number of terms

Example:

If the first term a = 3, the last term l = 15, and the number of terms n = 6, then the sum of the first 6 terms is:

Sn = 6/2 [3 + 15] = 3 x 18 = 54

Therefore, the sum of the first 6 terms is 54.

➱ Sum of First 'n' Natural Numbers
Formula:
Sn = n(n + 1) / 2
➱ Arithmetic Mean

If a, b, c are in A.P., then the arithmetic mean of a & c is given by:

b = (a + c) / 2
Example:
For the A.P. 1, 4, 7, 10, 13, ..., the arithmetic mean of 4 and 10 is:
(4 + 10) / 2 = 7

© CBSE Board Mathematics Notes | By Pratap Sanjay Sir