The common difference in the arithmetic sequence 7, 3, –1, –5, … is –4.

In any **arithmetic** **sequence**, the **common difference** is obtained by subtracting a term from the subsequent term. For clarity, if we take the second term, 3, and subtract the first term, 7, we get 3 – 7 = –4. If I subtract any term from the term that follows it in this **sequence**, the result will consistently be –4.

This value is what allows us to predict the pattern of the **sequence**, making it a foundational aspect of understanding **arithmetic sequences**.

Each term in an **arithmetic** **sequence** can be found using the formula $A_n = A_1 + (n – 1)d$, where $A_1$ is the first term, $A_n$ is the nth term, and ( d ) is a **common** **difference**. In this **sequence**, $A_1 = 7$ and ( d = –4 ), confirm that any term can be calculated by starting at 7 and adding the product of the position of the term minus one and –4.

## Calculating Common Difference

In an **arithmetic** **sequence**, the **common** **difference** is the constant value that each term is increased or decreased by to get the next term. It’s the key to understanding and working with these sequences. To find it, simply subtract any term from the following term. In our sequence, let’s look at the first two terms:

- To find the common difference
**(d)**, subtract the second term**(a₂)**from the first term**(a₁)**: $$ d = a₂ – a₁ $$ - For our sequence: $$ d = 3 – 7 $$ $$ d = -4 $$

This calculation shows that the **common** **difference** is **-4**, which is a negative value, indicating each term in the **sequence** decreases by 4 from its predecessor.

To confirm our finding, let’s subtract consecutive terms throughout the **sequence**:

Term Being Subtracted (n) | Resulting Value |
---|---|

3 (a₂) – 1 (a₃) | -4 |

-1 (a₃) – -5 (a₄) | -4 |

Here, we consistently see that **subtracting** any term from its subsequent term yields **-4**, reiterating our **calculation** is correct. It’s crucial to check more than one pair of terms to ensure that the **difference** remains constant throughout the **sequence**.

Now, if I were to use a calculator designed for **arithmetic** **sequences**, I would input **7** as the first term, and a **negative number, -4,** as the **common** **difference**. The calculator would then confirm the subsequent terms or even the value of the n-th term based on this **common** **difference**.

We’ve established that in our **arithmetic sequence**, the **common** **difference** is **-4**. This value helps us understand the behavior of the **sequence** and predict future terms.

## Conclusion

In the **arithmetic** **sequence** given, each number decreases by 4 from the previous one. This **characteristic** is defined as the **common** **difference**, typically represented by the symbol ( d ). Here, we calculate the common difference by subtracting any term from the one that follows it; for example, ( 3 – 7 = -4 ). To check consistency, a subsequent pair such as ( -1 – 3 ) also equals ( -4 ), confirming that ( d = -4 ).

**Arithmetic** **sequences** follow a pattern where each term is derived by adding the **common** **difference** to the preceding term. In mathematical terms, if $a_n$ represents the $n^{th}$ term, and $a_1$ is the first term, then the $n^{th}$ term is found using the formula:

$a_n = a_1 + (n-1)d$

Applying this to our **sequence** allows us to find any term’s value. **Arithmetic** **sequences** like this one are linear, meaning each term increases or decreases at a constant rate. Recognizing this rate, the **common difference** is crucial in understanding the structure of the sequence and **predicting subsequent** terms.

Here, the **subsequent** term after -5 would be ( -5 – 4 = -9 ), following the established **pattern**. Keep this simple process in mind to identify the **common** **difference** in any **arithmetic** **sequence** you may encounter.