Find 4 arithmetic means between 3 and 18.

Explanation

To find 4 arithmetic means between 3 and 18, we can follow these steps:

### Step 1: Identify the first and last terms
The first term (A1) is 3 and the last term (A6) is 18. We need to find the 4 arithmetic means, which are A2, A3, A4, and A5.

### Step 2: Determine the number of terms
We have a total of 6 terms in this sequence:
- A1 = 3 (first term)
- A2 (first arithmetic mean)
- A3 (second arithmetic mean)
- A4 (third arithmetic mean)
- A5 (fourth arithmetic mean)
- A6 = 18 (last term)

### Step 3: Use the nth term formula for arithmetic sequences
The nth term of an arithmetic sequence can be expressed as:
\[ A_n = A + (n-1)D \]
where:
- \( A \) is the first term,
- \( D \) is the common difference,
- \( n \) is the term number.

For our case, we want to find A6:
\[ A_6 = A + 5D \]
Substituting the known values:
\[ 18 = 3 + 5D \]

### Step 4: Solve for the common difference (D)
Rearranging the equation:
\[ 5D = 18 - 3 \]
\[ 5D = 15 \]
\[ D = \frac{15}{5} = 3 \]

### Step 5: Calculate the arithmetic means
Now that we have the common difference \( D = 3 \), we can find the arithmetic means:
- A2 = A1 + D = 3 + 3 = 6
- A3 = A2 + D = 6 + 3 = 9
- A4 = A3 + D = 9 + 3 = 12
- A5 = A4 + D = 12 + 3 = 15

### Step 6: List the arithmetic means
The 4 arithmetic means between 3 and 18 are:
- A2 = 6
- A3 = 9
- A4 = 12
- A5 = 15

### Final Answer
The four arithmetic means between 3 and 18 are 6, 9, 12, and 15.

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