The first term and the sixth term of an arithmetic sequence are 8 and 3, respectively. Find the common difference.


To find the common difference \(d\) in an arithmetic sequence, we can use the formula for the \(n\)-th term of the sequence:

\[
a_n = a_1 + (n-1) \cdot d
\]

Here:
- \(a_1 = 8\) (the first term)
- \(a_6 = 3\) (the sixth term)
- \(n = 6\)

Substitute these values into the formula for \(a_n\):

\[
a_6 = a_1 + (6-1) \cdot d
\]

\[
3 = 8 + 5d
\]

Now, solve for \(d\):

\[
3 - 8 = 5d
\]

\[
-5 = 5d
\]

\[
d = \frac{-5}{5} = -1
\]

So, the common difference \(d\) is \(-1\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the \(n\)-th term of an arithmetic sequence given \(a_1\) and \(d\)?
2. What is the sum of the first 10 terms of this arithmetic sequence?
3. If the common difference was positive, what would \(a_6\) be?
4. How would the sequence change if the first term were negative?
5. What is the general formula for the sum of an arithmetic sequence?

**Tip:** In arithmetic sequences, the difference between consecutive terms is always constant, which makes it easy to identify patterns or missing terms.

The first term and the sixth term of an arithmetic sequence are 8 and 3, respectively. Find the common difference.

Discover how to find the common difference in an arithmetic sequence when the first term is 8 and the sixth term is 3. This problem involves key algebraic concepts, including the formula for the n-th term of an arithmetic sequence. Ideal for students in Grades 6-8.

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