A sequence is a set of numbers that are in a particular order is called sequence.
Example (1): 1, 3, 5, 7, 9, .......
Example (2): 2, 4, 6, 8, 10, .......
When the terms of a sequence are written with the addition or subtraction operation then it is called a series.
Example (1): 1 + 3 + 5 + 7 + 9 + .......
Example (2): 1 - 3 - 5 - 7 - 9 - .......
Example (3): 2 + 4 + 6 + 8 + 10 + .......
A sequence of numbers in which each term is driven from the preceding term by adding or subtracting a fixed number is called arithmetic progression and the fixed number which is added or subtracted by each term is called "common difference" and it is denoted by "d". The common difference can be positive, negative or zero.
Example (1): 1, 3, 5, 7, 9, ......
Here 2 is added with each term to get the next term so 2 is a common difference (d).
Example (2): 2, 3, 4, 5, 6, ......
Here 1 is added with each term to get the next term so 1 is a common difference (d).
Let's consider the first term of a series is "a" and the common difference is "d" then the general term of arithmetic progression will be
a, a+d, a+2d, a+3d, a+4d, .....
Here the common difference (d) in the second term is multiplied by 1, d in the third term is multiplied by 2, d in the fourth term is multiplied by 3 and so on.
Hence it can be written as
$$ n^{th} \ term \ (l) = a + (n - 1) \ d $$
Example: If the nth term of a series is (2n - 3) then find the 7th term of that series?
Solution: $$ n^{th} \ term = (2n - 3) $$ $$ 7^{th} \ term = (2 \times 7 - 3) $$ $$ = 14 - 3 $$ $$ = 11 $$ Hence 7th term of that series is 11.