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Let's solve each part of the activity by finding the arithmetic means as required:

### 1. Find the two arithmetic means between -4 and 5.

To insert two arithmetic means between -4 and 5, we need to find three terms: the first term, two arithmetic means, and the last term, forming an arithmetic sequence. The formula for the general term of an arithmetic sequence is:

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

Where:
- \(a_1 = -4\) (the first term),
- \(a_4 = 5\) (the fourth term, since there are two arithmetic means),
- \(d\) is the common difference,
- \(n = 4\) (the total number of terms).

Using the formula for the 4th term:
\[
5 = -4 + 3d
\]
\[
5 + 4 = 3d \quad \Rightarrow \quad 9 = 3d \quad \Rightarrow \quad d = 3
\]

Now, we can find the arithmetic means (the second and third terms):
- \(a_2 = a_1 + d = -4 + 3 = -1\),
- \(a_3 = a_1 + 2d = -4 + 6 = 2\).

Thus, the two arithmetic means are **-1** and **2**.

---

### 2. Find the three arithmetic means between 100 and 40.

Here, we need to insert three arithmetic means, meaning we have five terms in total: 100, three means, and 40.

Using the same approach:
- \(a_1 = 100\),
- \(a_5 = 40\),
- \(n = 5\).

For the 5th term:
\[
40 = 100 + 4d
\]
\[
40 - 100 = 4d \quad \Rightarrow \quad -60 = 4d \quad \Rightarrow \quad d = -15
\]

Now, calculate the arithmetic means:
- \(a_2 = a_1 + d = 100 - 15 = 85\),
- \(a_3 = a_1 + 2d = 100 - 30 = 70\),
- \(a_4 = a_1 + 3d = 100 - 45 = 55\).

Thus, the three arithmetic means are **85**, **70**, and **55**.

---

### 3. Find the three arithmetic means between 22 and 52.

Here, we have a similar situation as the previous one, with three arithmetic means, so five terms total.

- \(a_1 = 22\),
- \(a_5 = 52\),
- \(n = 5\).

For the 5th term:
\[
52 = 22 + 4d
\]
\[
52 - 22 = 4d \quad \Rightarrow \quad 30 = 4d \quad \Rightarrow \quad d = 7.5
\]

Now, calculate the arithmetic means:
- \(a_2 = a_1 + d = 22 + 7.5 = 29.5\),
- \(a_3 = a_1 + 2d = 22 + 15 = 37\),
- \(a_4 = a_1 + 3d = 22 + 22.5 = 44.5\).

Thus, the three arithmetic means are **29.5**, **37**, and **44.5**.

---

### 4. Find the two arithmetic means between 11 and 5.

Finally, we need two arithmetic means between 11 and 5, forming four terms in total.

- \(a_1 = 11\),
- \(a_4 = 5\),
- \(n = 4\).

For the 4th term:
\[
5 = 11 + 3d
\]
\[
5 - 11 = 3d \quad \Rightarrow \quad -6 = 3d \quad \Rightarrow \quad d = -2
\]

Now, calculate the arithmetic means:
- \(a_2 = a_1 + d = 11 - 2 = 9\),
- \(a_3 = a_1 + 2d = 11 - 4 = 7\).

Thus, the two arithmetic means are **9** and **7**.

---

Do you need any further clarifications or details on these calculations?

### Related Questions:
1. How do you find the common difference in any arithmetic sequence?
2. What happens if you insert more arithmetic means between two numbers?
3. How is the formula for the general term of an arithmetic sequence derived?
4. Can you explain how the concept of arithmetic means applies to real-life situations?
5. How would the approach change for finding arithmetic means between fractions?

### Tip:
When solving problems involving arithmetic sequences, always carefully label the number of terms in the sequence, as this helps ensure accurate setup of the equations.

Find the arithmetic means of the following sequence.
1. Find the two arithmetic means between -4 and 5.
2. Find the three arithmetic means between 100 and 40.
3. Find the three arithmetic means between 22 and 52.
4. Find the two arithmetic means between 11 and 5.

Solve problems involving finding arithmetic means in sequences. Learn how to calculate two arithmetic means between -4 and 5, three arithmetic means between 100 and 40, three arithmetic means between 22 and 52, and two arithmetic means between 11 and 5. Suitable for Grade 10 students.

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