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The question in the image reads:

"If seven AM (Arithmetic Means) are inserted between 7 and 27, then the fourth AM will be?"

This is a problem involving arithmetic progression (AP).

### Step-by-step solution:

1. The first term of the AP (\( a_1 \)) is 7, and the last term (\( a_9 \)) is 27, because there are seven arithmetic means inserted between them, resulting in a total of 9 terms.

2. The general form of an arithmetic progression is:
   \[
   a_n = a_1 + (n - 1)d
   \]
   where:
   - \( a_n \) is the nth term,
   - \( a_1 \) is the first term,
   - \( d \) is the common difference,
   - \( n \) is the term number.

3. Here, \( a_9 = 27 \), \( a_1 = 7 \), and \( n = 9 \). Using the formula for the nth term:
   \[
   a_9 = a_1 + 8d
   \]
   Substituting the known values:
   \[
   27 = 7 + 8d
   \]
   \[
   27 - 7 = 8d
   \]
   \[
   20 = 8d
   \]
   \[
   d = \frac{20}{8} = 2.5
   \]

4. Now, to find the fourth arithmetic mean (\( a_5 \)), we use the same formula:
   \[
   a_5 = a_1 + (5 - 1)d = 7 + 4 \times 2.5 = 7 + 10 = 17
   \]

Thus, the fourth arithmetic mean is **17**.

Would you like further explanation or details? Here are 5 follow-up questions related to this topic:

1. What is the general formula for finding any term in an arithmetic progression?
2. How do you calculate the sum of the first \( n \) terms of an arithmetic progression?
3. How can you determine the common difference if you're given the first and the last terms?
4. How does an arithmetic progression differ from a geometric progression?
5. Can you explain how the number of terms affects the common difference in an arithmetic progression?

**Tip**: Always double-check the number of terms and their positions when solving arithmetic progression problems, especially when inserting means between two numbers.

If seven AM are inserted between 7 and 27, then the fourth AM will be

Discover how to find the fourth arithmetic mean when seven arithmetic means are inserted between 7 and 27. This problem uses the arithmetic progression formula and explains the calculation of the common difference. Ideal for students in Grades 8-10.

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