7. The first term of an arithmetic sequence is 28 while the 12th term is 105. What is the common difference of the sequence
a. 3
b. 5
c. 6
d. 7

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

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

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

Given:
- The first term \(a_1 = 28\),
- The 12th term \(a_{12} = 105\),
- We want to find the common difference \(d\).

### Step-by-step Solution:

1. **Set up the formula for the 12th term**:

   \[
   a_{12} = a_1 + (12 - 1)d
   \]

2. **Substitute the known values into the formula**:

   \[
   105 = 28 + 11d
   \]

3. **Solve for \(d\)**:

   \[
   105 - 28 = 11d
   \]

   \[
   77 = 11d
   \]

   \[
   d = \frac{77}{11}
   \]

   \[
   d = 7
   \]

### Conclusion:

The common difference of the sequence is \(d = 7\).

Thus, the correct answer is **d. 7**.

Would you like more details or have any questions about this?

Here are 5 related questions to consider:

1. How would you find the 20th term of the sequence given the first term and common difference?
2. What is the sum of the first 15 terms of this arithmetic sequence?
3. How does changing the first term affect the arithmetic sequence?
4. What happens if the common difference is negative? How would that affect the sequence?
5. Can an arithmetic sequence have a non-integer common difference?

**Tip:** For any arithmetic sequence, the difference between consecutive terms is always constant, which makes it easy to predict future terms.

Learn how to find the common difference of an arithmetic sequence with the first term 28 and the 12th term 105. Step-by-step solution provided.

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