To understand the term ‘Arithmetic Sequence’, first, we must understand the sequence’s meaning.
Key Takeaways
- An arithmetic sequence is a sequence of numbers where each term is obtained by adding a constant value called the common difference to the previous term.
- The formula for the nth term of an arithmetic sequence is given by an = a1 + (n-1)d, where a1 is the first term and d is a common difference.
- Arithmetic sequences find wide applications in various fields, including physics, finance, and computer science.
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, we add the same value or term each time to 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, 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
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