How to use the Arithmetic Sequence Formula
What is Sequence?
Sequence means something which is arranged, just like a list of numbers. Below is the series of numbers separated by commas between each other.
17, 21, 25, 29, 33
A series is a sum you're adding together but a sequence is a list. Arithmetic means i.e. some common difference that you're adding to get to the next term and that thing that you're adding is called d, the common difference.
What is the Common Difference?
The common difference is the same thing that we are adding in the previous terms to get the next term of sequence. The common difference in mathematics is denoted by “d”. The common difference arithmetic sequence formula is:
d = an - an-1
Why do we call it the Common Difference?
Difference means subtract.
If you take 21 - 17 what do you get for?
If you take 25 - 21 what are you getting for?
If you take 29 - 25 what are you getting for?
So the common difference is actually what we're adding to get to the next term in the list or in the sequence. This is why we call it d.
Common Difference examples 1
If we subtract next term from the previous term in a sequence of numbers,
21 - 17 = 4
We get 4 as a common difference.
Common Difference examples 2
Similarly, If we subtract next term (25) from the previous term (21) in a sequence of numbers,
25 - 21 = 4
We again get 4 as a common difference.
If we have a sequence of numbers, The same thing is going through the whole sequence.
sequence = 17 , 21 , 25 , 29 , 33 ....
D = 4
Number of terms and Values in a Sequence
There are always different numbers of values and terms present in a sequence of numbers. In the given Sequence,
sequence = 17 , 21 , 25 , 29 , 33 ...
This sequence starts from 17. This is actually called n = 1, n = 2, n = 3 and so on. This means these numbers are the first term, second term and third term of the sequence.
Lets understand the values in a sequence of numbers.
a1 = 17
a2 = 21
a3 = 25
a4 = 29
a5 = 31
The above numbers indicate the value of the first term, second term, third term, fourth term & the fifth term.
As we mentioned above, n is the number of the term while a indicates the value of that term and what that term actually equals.
How to find the Next term of Sequence
Let's understand how we find out what the term is next to a5 = 33.
Just focus on the above values where a5 is the 5th term and we are going to find the value of n = 6. The next value will be the a6 as the 6th term.
Example:
We have the value of n = 6, let’s find the sixth term?
We will start off from n = 1 and we will add 4 (common difference).
We will add the common difference five times depending upon how many terms are already available in a sequence.
So mathematically, it will be written as:
a6 = 17 + 4(5)
This makes us able to find the sixth term i.e. a6. So to get to the sixth term we only have to add 4 one less time.
General formula Arithmetic Sequence
Now we will get the general formula for arithmetic sequence. Based on above scenario,
a6 = 17 + 4(5)
17 = a1
4 = d
And, 5 = n-1
Or, in simple we may write,
an = a1 + d(n-1)
So in this case we're getting
a6 = 17 + 4(5)
a6 = 37
n = 1 + 4 or 5 times.
The value will be 37. It is only because we are adding 4 each time. So now what we have is an explicit formula meaning that it will take you right to any particular term. So let's write this in a little bit more general way,
an = a1 + d(n-1)
Now in our case,
an = 17 + 4(n-1)
an = 17 + 4n - 4
an = 13 + 4n
So now we have 13 + 4n. This formula will allow us to find any term of this sequence.
Example:
With this approach and formula, we can get the hundredth term as well.
an = 13 + 4n
For n = 100,
a100 = 13 + 4(100)
A100 = 413
This is how we calculate arithmetic sequence formulas with this approach. We hope you liked our examples and it helped you in learning the formula.
Please provide your valuable feedback as well.
