Table of contents  
What is a Sequence ?  
Arithmetic Progression (A.P.)  
nth term of an A.P.  
Sum of n terms of an A.P. 
A sequence is an arrangement of numbers in a definite order and according to some rule.
Example:
The various numbers occurring in a sequence are called ‘terms’. Since the order of a sequence is fixed, therefore the terms are known by the position they occupy in the sequence.
Example: If the sequence is defined as
An Arithmetic progression is a special case of a sequence, where the difference between a term and its preceding term is always constant, known as common difference, i.e., d. The arithmetic progression is abbreviated as A.P.
The general form of an A.P. is
∴ a, a + d, a + 2d,… For example, 1, 9, 11, 13.., Here the common difference is 2. Hence it is an A.P.
In an A.P. with first term a and common difference d, the nth term (or the general term) is given by .
a_{n} = a + (n – 1)d.
where [a = first term, d = common difference, n = term number
For Example,
To find seventh term put n = 7
∴ a_{7} = a + (7 – 1)d or a_{7} = a + 6d
Example 1: Find the value of n, if a = 10, d = 5, an = 95.
Sol: Given, a = 10, d = 5, an = 95
From the formula of general term, we have:
a_{n} = a + (n − 1) × d
95 = 10 + (n − 1) × 5
(n − 1) × 5 = 95 – 10 = 85
(n − 1) = 85/ 5
(n − 1) = 17
n = 17 + 1
n = 18
Example 2: Find the 20th term for the given AP:3, 5, 7, 9, ……
Sol: Given,
3, 5, 7, 9, ……
a = 3, d = 5 – 3 = 2, n = 20
a_{n} = a + (n − 1) × d
a_{20} = 3 + (20 − 1) × 2
a_{20} = 3 + 38
⇒a_{20} = 41
The sum of the first n terms of an A.P. is given by
where [a = first term, d = common difference, n = term number
Example: Find the sum of the first 30 multiples of 4.
Sol: The first 30 multiples of 4 are: 4, 8, 12, ….., 120
Here, a = 4, n = 30, d = 4
We know,
S30 = n/2 [2a + (n − 1) × d]
S30 = 30/2[2 (4) + (30 − 1) × 4]
S30 = 15[8 + 116]
S30 = 1860
Note: If a, b, c are in A.P. then b = and b is called the arithmetic mean of a and c.
116 videos420 docs77 tests

1. What is an Arithmetic Progression (A.P.)? 
2. How do you find the nth term of an Arithmetic Progression (A.P.)? 
3. What is the formula for the sum of n terms of an Arithmetic Progression (A.P.)? 
4. How can you identify if a given sequence is an Arithmetic Progression (A.P.)? 
5. Can the common difference in an Arithmetic Progression (A.P.) be negative? 

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