What is an Arithmetic Sequence?

For understanding the term ‘Arithmetic Sequence’, first we have to understand what is the meaning if sequence.

Sequence

A sequence is a group of numbers, which are in order. For example; 3,5,7,9… and so on.

Each number in the sequence or group of numbers is called a term. Sometimes they are called “elements” or “members”. Now,

What is Arithmetic Sequence?

In this sequence, the difference between one term and the next follows a constant behaviour. In other words, in here we add the same value or term each time to the infinity.

Example:
1,4,7,13,16,19,20,25,… here, this sequence follows the difference of 3 between numbers. The pattern is continuous by adding three each time as shown below,

So, commonly we write a correct sequence like this, or the formula for the correct sequence is;

{a, a+d, a+2d, a+3d, …}

In here,

  • ‘a’ represents the first term of the sequence, and
  • ‘d’ represents the difference between the terms, which is called the (common difference) of the sequence.

Example: (Continued from above)

1,4,7,13,16,19,20,25,…

It has,

  • ‘a’ = 1 (which is the 1st term)
  • ‘d’ = 3 (which is the “common difference” between the terms)

We get,

Formula is : { a, a+d, a+2d, a+3d,…}
{ 1, 1+3, 1+2×3, 1+3×3,…}
{1,4,7,10,…}

Rule

We can also write ‘AS’ (Arithmetic Sequence) as a rule,

Xn = a + d(n-1)
We use “n-1” because in the first term the ‘d’ is not used

Example : Find the 9th term from this sequence.


3, 8, 13, 18, 23, 28, 33, 38, …

Now, this sequence here has a common difference of 5 between them.

The value of d and a are:

  • d = 5 (the common difference between the terms)
  • a = 3 (the first term of the sequence)

Now, by using the formula,

Xn = a + d(n-1)
= 3 + 5(n-1)
= 3 + 5n – 5
= 5n – 2

hence, the 9th term is. Here, n = 9.


X9 = 5 x 9 – 2
= 43

References

  1. https://pdfs.semanticscholar.org/a318/b30ce0239c43731610f354cdc7ad500eb77b.pdf
  2. https://www.sciencedirect.com/science/article/pii/S0096300308008837
  3. https://repository.unej.ac.id/handle/123456789/98520
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