Find three arithmetic means between 3 and 19.

Answer

To find three arithmetic means between 3 and 19, we can follow these steps:

### Step 1: Understand the concept of arithmetic means
An arithmetic mean is a number that is equally spaced between two other numbers. If we have two numbers \( a \) and \( b \), and we want to find \( n \) arithmetic means between them, we can express them as:
- First term: \( a \)
- Second term: \( a + d \)
- Third term: \( a + 2d \)
- ...
- Last term: \( b \)

Where \( d \) is the common difference.

### Step 2: Identify the known values
In this case, we have:
- First term \( a = 3 \)
- Last term \( b = 19 \)
- Number of arithmetic means \( n = 3 \)

### Step 3: Calculate the total number of terms
The total number of terms in the sequence will be:
\[ n + 2 = 3 + 2 = 5 \]
So, we have 5 terms in total: \( 3, m_1, m_2, m_3, 19 \).

### Step 4: Find the common difference \( d \)
The formula for the \( n \)-th term of an arithmetic sequence is given by:
\[ b = a + (n + 1)d \]
Substituting the known values:
\[ 19 = 3 + (5 - 1)d \]
This simplifies to:
\[ 19 = 3 + 4d \]
Subtracting 3 from both sides gives:
\[ 16 = 4d \]
Dividing both sides by 4 gives:
\[ d = 4 \]

### Step 5: Calculate the arithmetic means
Now we can find the three arithmetic means:
1. First mean: \( m_1 = a + d = 3 + 4 = 7 \)
2. Second mean: \( m_2 = a + 2d = 3 + 2(4) = 3 + 8 = 11 \)
3. Third mean: \( m_3 = a + 3d = 3 + 3(4) = 3 + 12 = 15 \)

### Step 6: Write the final answer
The three arithmetic means between 3 and 19 are:
\[ 7, 11, 15 \]

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