What is an arithmetic succession
Succession is an ordered sequence of numbers, such as:
A succession of real numbers is an application of the set N (set of natural number excluding zero) in the set R of the real numbers.
It is called the term of succession to each one of the elements that make un the succession. To represent the different terms of a sequence, the same letter is used with different subscripts, which indicate the place which that therm occupies in the sequence.
- In the succession: a) 1, 2, 3, 4, 5, 6,… we have that: a5 = 5, because it is the term of the succession that occupies the fifth place.
- In the succession: b) 2, 4, 6, 8 , 10,… the third term would be written by b3 and it would corresponded to the value 6.
The really important thing when it comes to naming the terms of a sequence is the subscript because it denotes the place it occupies in the sequence. The letters with which the succession is designated are different for different sequences and are usually lowercase letters.
We called the general term of succession to the term that occupies the n-th and it is written with the letter that denotes the succession (for example a) with subscript n:(an).
If we focus on the values taken by the subscripts, we see that they are natural numbers, but the terms of the sequence do not have to be so, that is, the values taken by the sequence are real numbers. Therefore, we can affirm that a succession of real numbers is an application that corresponds to each natural number a real number.
Ways to define a succession
There are several ways to define a succession:
- Giving its general term: A sequence has infinito terms and is frequently expressed by its general term an, that since an is a function that depends on n, it is enough to give natural values to the indeterminate n to obtain any term of the sequence.
For example, the sequence has as a general rule:
- Giving a property that meets the terms of the succession: If we ensure a property that meets the succession, it is easier to know the value of an.
– Ex: thanks to the following properties, get the successions:
- A succession of pair numbers: 2, 4, 6, 8, …
- A succession of prime numbers: 2, 3, 5, 7, 11, …
- A succession of the natural numbers ending in 9: 9, 19, 29, 39, …
- By a law of recurrence: you can get a term from previous ones. – Ex: Knowing that the first term of succession is 2, and each term, except the first one, is three times the previous term, write the first terms of the sequence.
It is said that a succession of real numbers is increasing when it is verified that each term is less than or equal to the next one, that is:
On the other side, it is said that a succession is decreasing if each term is greater than or equal to the next one: