Insert 3 arithmetic means between 2 and 10.

Answer

To insert 3 arithmetic means between 2 and 10, we can follow these steps:

### Step 1: Identify the first and last terms
We have the first term \( a = 2 \) and the last term \( a_5 = 10 \). We need to insert 3 arithmetic means between them, which will give us a total of 5 terms in the arithmetic progression (AP).

### Step 2: Use the formula for the nth term of an AP
The formula for the nth term of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

### Step 3: Set up the equation for the last term
For our case, we know:
\[ a_5 = 10 = 2 + (5 - 1) \cdot d \]
This simplifies to:
\[ 10 = 2 + 4d \]

### Step 4: Solve for the common difference \( d \)
Subtract 2 from both sides:
\[ 10 - 2 = 4d \implies 8 = 4d \]
Now, divide both sides by 4:
\[ d = \frac{8}{4} = 2 \]

### Step 5: Find the arithmetic means
Now that we have the common difference \( d = 2 \), we can find the three arithmetic means.

1. **First Arithmetic Mean (AM1)**:
\[ AM1 = a + d = 2 + 2 = 4 \]

2. **Second Arithmetic Mean (AM2)**:
\[ AM2 = a + 2d = 2 + 2 \cdot 2 = 2 + 4 = 6 \]

3. **Third Arithmetic Mean (AM3)**:
\[ AM3 = a + 3d = 2 + 3 \cdot 2 = 2 + 6 = 8 \]

### Step 6: Write the complete arithmetic progression
The complete arithmetic progression with the 3 means inserted is:
\[ 2, 4, 6, 8, 10 \]

### Final Answer
The three arithmetic means inserted between 2 and 10 are 4, 6, and 8.

---