Sequence Series is a very important part of mathematics. When we write numbers in an order then we call it sequence of numbers.
The list of numbers follows a particular trend in that. The numbers in a seq are called terms of the seq. We cannot flip the terms in a seq.
Their order cannot be changed. We do the counting of terms from left to right.
For example: - 1, 3, 5, 7, 9 is a seq of five terms where first term is 1 and third term is 5 and last term is 7.
If we look at this seq we see that this seq follow a certain trend and that is every next term is two more than the previous term. Now what is a series?
A series is when we add the terms of a seq that constitutes to form a series. There are different types of seq and ser that exists and they are: -
1. Arithmetic Sequences and Series– In this type of seq and ser, the next term can be determined by adding the common difference to the previous term. The general arithmetic seq and ser is given as a, a+d, a+2d, a+3d, a+4d… where ‘a’ is the first term and d is the common difference. For example: - 1, 4, 7, 10, 13, 16. Here we see that first term is 1 and the common difference is 3
2. Geometric Sequences and Series– In this type of seq and ser, the next term can be determined by multiplying the common multiple by the previous term. The general geometric seq and ser is given as a, ar^2, ar^3, ar^4… where ‘a’ is the first term and r is the common multiple. For example: - 2, 6, 18, 54. Here we see that the first term is 2 and the common multiple is 3 which can be determined by dividing any term by its previous term.
3. Infinite Sequences and Series– In this type of seq and ser, the last term is unknown to us. We do not have fixed number of terms in that as it is an infinite ser. For example: - 2, 6, 10, 14…..
Series and Sequences Formulas are used to determine the first term, last term, sum of terms, number of terms or the common difference or ratio.