A sequence is basically a list of numbers written in a designated forum. For e.g. we list a few dates of our friends’ birthdays, 10,13,21,23,29. This complete list of the numbers is called a sequence. Let us have a look into another sequence. For e.g 2,4,6,8,10,12,14,16… is an example of an infinite sequence. The infinite sequence doesn’t have any endpoints and it goes on and on. Thus an infinite sequence is represented by the three dots after the list of numbers which denotes that the numbers are never-ending. Similarly, a finite sequence can be demonstrated with the example of friends’ birthday dates as shown above.
Generally, each number within the sequence is commonly known as a term. The first term is represented by the first number, the second term as the second number, and so forth. A sequence can be formed by any list of numbers irrespective of any pattern. Moreover, a sequence can also consist of words or alphabets. It is not necessary that the list would contain only numbers. To show you with an example: {bat, square, elephant} can also be termed as a sequence. Let us check what an arithmetic sequence would be like.
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What is an Arithmetic Sequence?
An arithmetic sequence is a sequence where the same number is added or subtracted to get the next number in the list. For example, there is a sequence: 1, 3, 5, 7, 9… which is known to be an arithmetic sequence since 2 is added every time to get the next number. Similarly, the sequence 30, 28, 26, 24, 22… is also an arithmetic sequence where 2 is subtracted each time to get the next possible number. There is also another process to check an arithmetic sequence. The difference between the adjacent numbers should be a constant. For example: (3 – 1 =2), (5 – 3 =2), (7 – 5 = 2) and so on. Thus the constant here is 2 as the difference between two adjacent numbers comes up as 2.
Thus the difference here to get the next possible number is called a common difference. In the above example, 2 is the common difference to get the next number by adding it to the previous one. A common difference is generally denoted by ‘d’. Thus if a common difference of an arithmetic sequence is 5, that means that 5 is added each time to get the next number within the sequence. Similarly, if the common difference within an arithmetic sequence is -3, that means 3 is being subtracted to get the next number in the sequence.
Labeling the terms of a sequence
Let us find an example of a sequence which is as 3, 5,7, 9, 11… Here the first term is 3, the second is 5, and so on. So with respect to formulae, how do we label these terms? We, in general, use the letter ‘a’ to denote the number with a subscript below to denote the term. So for the above example, 3 is the first number and can be denoted as a1 which resembles the first term. Similarly, for the other numbers, the notations are utilized such as a2 = 5, a3 = 7, a4 = 9, a5 = 11 for the sequence present. Sometimes we can see a never-ending sequence with the term n. It basically looks like an. This is the nth term and can be any number within the sequence.
General Formula of an Arithmetic Sequence
The general formula to denote an Arithmetic Sequence is:
an = a1 + d(n – 1)
Thus to find the nth term of the sequence, we need to add the first term with the common difference n – 1 time to get the value of the nth number. That means to get the 30th term we need to add the first term and the common difference 29 times. The reason why we use n-1 is that the first number doesn’t have any common difference.
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