Which of the following is an arithmetic sequence? That said, it is the option in which the difference between every pair of consecutive terms is constant. To answer this type of question correctly, subtract each term from the term that follows it and check whether the same value appears throughout the entire sequence Most people skip this — try not to..
Introduction
Questions such as “Which of the following is an arithmetic sequence?” commonly appear in algebra assessments, including Apex-style practice problems. These questions test whether you can recognize a pattern and determine whether it follows the rules of an arithmetic sequence Most people skip this — try not to..
An arithmetic sequence is an ordered list of numbers in which each term changes by the same amount. As an example, in the sequence 3, 7, 11, 15, each term increases by 4. That fixed amount is called the common difference. Because the difference remains constant, the sequence is arithmetic.
The key to solving these questions is not simply looking for a pattern. It is verifying that every consecutive difference is identical.
What Is an Arithmetic Sequence?
An arithmetic sequence is a sequence whose terms increase or decrease by a fixed number. This fixed number is known as the common difference, usually represented by the letter (d).
If the sequence is written as:
[ a_1, a_2, a_3, a_4, \ldots ]
then the common difference can be found using:
[ d = a_n - a_{n-1} ]
Basically, each term minus the previous term must equal the same value.
For example:
[ 4, 9, 14, 19, 24 ]
The differences are:
- (9 - 4 = 5)
- (14 - 9 = 5)
- (19 - 14 = 5)
- (24 - 19 = 5)
Since every difference is 5, the common difference is 5. So, the sequence is arithmetic And that's really what it comes down to..
A sequence can also decrease by a constant amount. For example:
[ 20, 15, 10, 5 ]
The differences are:
- (15 - 20 = -5)
- (10 - 15 = -5)
- (5 - 10 = -5)
The common difference is (-5), so this sequence is also arithmetic But it adds up..
How to Identify an Arithmetic Sequence
When asked which of the following is an arithmetic sequence, use the following steps It's one of those things that adds up..
1. Look at the sequence in its given order
The order of the terms matters. Do not rearrange the numbers unless the question specifically tells you to do so.
Take this: the numbers 2, 5, 8, and 11 form an arithmetic sequence when arranged as:
[ 2, 5, 8, 11 ]
Even so, if they appear as:
[ 2, 8, 5, 11 ]
they do not form an arithmetic sequence in that order Worth keeping that in mind..
2. Subtract each term from the next term
Find the difference between consecutive terms:
[ a_2 - a_1,\quad a_3 - a_2,\quad a_4 - a_3 ]
If all of these differences are equal, the sequence is arithmetic.
3. Check the entire sequence
It is not enough to check only the first two differences. You must verify that the common difference continues throughout the sequence.
For example:
[ 6, 10, 14, 20 ]
The first differences are:
- (10 - 6 = 4)
- (14 - 10 = 4)
These differences are equal, but the next difference is:
- (20 - 14 = 6)
Because the difference changes from 4 to 6, this sequence is not arithmetic It's one of those things that adds up..
4. Watch for negative common differences
An arithmetic sequence does not have to increase. It may decrease.
For example:
[ 12, 9, 6, 3 ]
has a common difference of (-3). This is arithmetic even though the numbers are getting smaller.
5. Remember that zero may be a common difference
A sequence such as:
[ 7, 7, 7, 7 ]
has a common difference of 0. Because
