How to Find the Common Difference of an Arithmetic Sequence
An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is a constant. This constant difference is known as the common difference. Finding the common difference of an arithmetic sequence is essential in various mathematical problems, such as determining the nth term, the sum of the first n terms, or identifying the general term of the sequence. In this article, we will discuss different methods to find the common difference of an arithmetic sequence.
Method 1: Using the Formula
The most straightforward method to find the common difference of an arithmetic sequence is by using the formula:
Common Difference (d) = a2 – a1
where a1 is the first term, and a2 is the second term of the sequence. By subtracting the first term from the second term, you can easily determine the common difference.
For example, consider the arithmetic sequence 3, 6, 9, 12, 15, …
In this sequence, the first term (a1) is 3, and the second term (a2) is 6. To find the common difference (d), we subtract a1 from a2:
d = a2 – a1
d = 6 – 3
d = 3
So, the common difference of this arithmetic sequence is 3.
Method 2: Using the General Term Formula
The general term formula for an arithmetic sequence is given by:
an = a1 + (n – 1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
By rearranging the formula, we can find the common difference:
d = (an – a1) / (n – 1)
This method is useful when you have the nth term and the first term of the arithmetic sequence.
For example, consider the arithmetic sequence 2, 5, 8, 11, 14, …
If we want to find the common difference using the general term formula, we can take the nth term (an) as 14 and the first term (a1) as 2. Let’s assume the term number (n) is 5.
d = (an – a1) / (n – 1)
d = (14 – 2) / (5 – 1)
d = 12 / 4
d = 3
Therefore, the common difference of this arithmetic sequence is 3.
Method 3: Using the Sum Formula
The sum formula for an arithmetic sequence is given by:
Sn = n/2 (a1 + an)
where Sn is the sum of the first n terms, a1 is the first term, an is the nth term, and n is the number of terms.
By rearranging the formula, we can find the common difference:
d = (2 Sn) / (n^2 – n)
This method is useful when you have the sum of the first n terms and the number of terms in the arithmetic sequence.
For example, consider the arithmetic sequence 1, 4, 7, 10, 13, …
If we want to find the common difference using the sum formula, we can take the sum of the first n terms (Sn) as 55 and the number of terms (n) as 5.
d = (2 Sn) / (n^2 – n)
d = (2 55) / (5^2 – 5)
d = 110 / (25 – 5)
d = 110 / 20
d = 5.5
Thus, the common difference of this arithmetic sequence is 5.5.
In conclusion, finding the common difference of an arithmetic sequence can be done using various methods, such as the formula, the general term formula, and the sum formula. By applying these methods, you can determine the common difference with ease and solve a variety of mathematical problems involving arithmetic sequences.
