Find the common difference of the arithmetic sequence with a4=10 and a11=45

To find the common difference of an arithmetic sequence, we use the formula for the nth term of an arithmetic sequence:

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

where:
- \(a_n\) is the nth term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.

We are given:
- \(a_4 = 10\),
- \(a_{11} = 45\).

Using the formula for both the 4th and 11th terms, we have:

\[
a_4 = a_1 + 3d = 10 \tag{1}
\]
\[
a_{11} = a_1 + 10d = 45 \tag{2}
\]

Now, subtract equation (1) from equation (2) to eliminate \(a_1\) and solve for \(d\):

\[
(a_1 + 10d) - (a_1 + 3d) = 45 - 10
\]

Simplifying:

\[
7d = 35
\]

\[
d = \frac{35}{7} = 5
\]

So, the common difference \(d\) is 5.

Do you want further details on the steps or have any other questions?

Here are 5 related questions:
1. How do you find the first term \(a_1\) in this sequence?
2. How do you find the nth term of an arithmetic sequence?
3. What is the sum of the first 11 terms in this sequence?
4. How would you determine the common ratio of a geometric sequence?
5. How do arithmetic sequences relate to linear equations?

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

Learn how to find the common difference of an arithmetic sequence where a4=10 and a11=45. This problem covers key algebraic concepts like arithmetic sequences and the formula for the nth term. Suitable for students in Grades 8-10.

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