Find 7 arithmetic means between 6 and 46.

Explanation

To find 7 arithmetic means between 6 and 46, we can follow these steps:

### Step 1: Identify the first and last terms
The first term (a) is 6 and the last term (a9) is 46. Since we need to find 7 arithmetic means, we will have a total of 9 terms in the sequence (the first term, the 7 means, and the last term).

### Step 2: Determine the number of terms
We have:
- First term (a1) = 6
- Last term (a9) = 46
- Total terms = 9 (1st term + 7 means + 1 last term)

### Step 3: Use the formula for the nth term of an arithmetic progression (AP)
The formula for the nth term of an AP is given by:
\[ a_n = a + (n - 1) \cdot d \]
Where:
- \( a \) = first term
- \( d \) = common difference
- \( n \) = term number

For our case:
\[ a_9 = a + (9 - 1) \cdot d \]
\[ 46 = 6 + 8d \]

### Step 4: Solve for the common difference (d)
Rearranging the equation:
\[ 46 - 6 = 8d \]
\[ 40 = 8d \]
\[ d = \frac{40}{8} = 5 \]

### Step 5: List the terms of the AP
Now that we have the common difference (d = 5), we can find the 7 arithmetic means:
- \( a_1 = 6 \)
- \( a_2 = a_1 + d = 6 + 5 = 11 \)
- \( a_3 = a_2 + d = 11 + 5 = 16 \)
- \( a_4 = a_3 + d = 16 + 5 = 21 \)
- \( a_5 = a_4 + d = 21 + 5 = 26 \)
- \( a_6 = a_5 + d = 26 + 5 = 31 \)
- \( a_7 = a_6 + d = 31 + 5 = 36 \)
- \( a_8 = a_7 + d = 36 + 5 = 41 \)
- \( a_9 = 46 \) (the last term)

### Step 6: Write the final sequence
The 7 arithmetic means between 6 and 46 are:
- 11, 16, 21, 26, 31, 36, 41

### Summary
The arithmetic means between 6 and 46 are: **11, 16, 21, 26, 31, 36, 41**.

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