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 (a) is 3, and the last term (l) is 18.

### Step 2: Determine the number of terms
We need to find 4 arithmetic means between 3 and 18. This means we have a total of 6 terms (3, a1, a2, a3, a4, 18).

### Step 3: Use the formula for the nth term of an arithmetic progression (AP)
The nth term of an AP 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, since we have 6 terms, we can express the 6th term (which is 18) as:
\[ a_6 = a + (6-1)d \]
\[ 18 = 3 + 5d \]

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

### Step 5: Calculate the arithmetic means
Now that we have the common difference \( d = 3 \), we can find the 4 arithmetic means:
- The second term (a1):
\[ a_1 = a + (2-1)d = 3 + 1 \cdot 3 = 3 + 3 = 6 \]

- The third term (a2):
\[ a_2 = a + (3-1)d = 3 + 2 \cdot 3 = 3 + 6 = 9 \]

- The fourth term (a3):
\[ a_3 = a + (4-1)d = 3 + 3 \cdot 3 = 3 + 9 = 12 \]

- The fifth term (a4):
\[ a_4 = a + (5-1)d = 3 + 4 \cdot 3 = 3 + 12 = 15 \]

### Step 6: List the arithmetic means
The 4 arithmetic means between 3 and 18 are:
\[ 6, 9, 12, 15 \]

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

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