Question
How to find the common difference of an arithmetic sequence represented by the number of squares in each term
Answers
GPT-4o mini
To find the common difference of an arithmetic sequence, you follow a simple process. An arithmetic sequence is defined by a set of numbers in which the difference between consecutive terms remains constant. This constant difference is called the common difference.
Here are the steps to find the common difference:
1. **Identify the Terms**: Write down the terms of the arithmetic sequence. For example, if the sequence is given as \( a_1, a_2, a_3, \ldots, a_n \), list out the numbers.
2. **Select Two Consecutive Terms**: Choose any pair of consecutive terms from the sequence. Typically, it's easier to work with the first two terms: \( a_1 \) and \( a_2 \).
3. **Calculate the Common Difference**: Use the formula:
\[
d = a_2 - a_1
\]
where \( d \) represents the common difference.
4. **Verify**: If desired, you can calculate the difference using other consecutive pairs (like \( a_3 - a_2 \) or \( a_4 - a_3 \)) to ensure that the difference is consistent across the sequence.
### Example
Suppose the arithmetic sequence is \( 4, 8, 12, 16, \ldots \).
1. Identify the terms: \( 4, 8, 12, 16 \).
2. Select the first two terms: \( a_1 = 4 \) and \( a_2 = 8 \).
3. Calculate the common difference:
\[
d = a_2 - a_1 = 8 - 4 = 4
\]
4. Verify with the next pair:
\[
a_3 - a_2 = 12 - 8 = 4
\]
\[
a_4 - a_3 = 16 - 12 = 4
\]
Since the common difference is consistent, you can conclude that the common difference is \( 4 \).
In summary, the common difference in an arithmetic sequence can be determined by subtracting any term from the next term, and it should yield the same result for any two consecutive terms in the sequence.
Here are the steps to find the common difference:
1. **Identify the Terms**: Write down the terms of the arithmetic sequence. For example, if the sequence is given as \( a_1, a_2, a_3, \ldots, a_n \), list out the numbers.
2. **Select Two Consecutive Terms**: Choose any pair of consecutive terms from the sequence. Typically, it's easier to work with the first two terms: \( a_1 \) and \( a_2 \).
3. **Calculate the Common Difference**: Use the formula:
\[
d = a_2 - a_1
\]
where \( d \) represents the common difference.
4. **Verify**: If desired, you can calculate the difference using other consecutive pairs (like \( a_3 - a_2 \) or \( a_4 - a_3 \)) to ensure that the difference is consistent across the sequence.
### Example
Suppose the arithmetic sequence is \( 4, 8, 12, 16, \ldots \).
1. Identify the terms: \( 4, 8, 12, 16 \).
2. Select the first two terms: \( a_1 = 4 \) and \( a_2 = 8 \).
3. Calculate the common difference:
\[
d = a_2 - a_1 = 8 - 4 = 4
\]
4. Verify with the next pair:
\[
a_3 - a_2 = 12 - 8 = 4
\]
\[
a_4 - a_3 = 16 - 12 = 4
\]
Since the common difference is consistent, you can conclude that the common difference is \( 4 \).
In summary, the common difference in an arithmetic sequence can be determined by subtracting any term from the next term, and it should yield the same result for any two consecutive terms in the sequence.