If the mean of 1002, 1004, 1006, 1008, 1010 is `(n)^3 + 6`, then find the value of n.

Answer

To find the value of \( n \) in the equation where the mean of the numbers 1002, 1004, 1006, 1008, and 1010 is equal to \( n^3 + 6 \), we can follow these steps:

### Step 1: Calculate the sum of the numbers
First, we need to find the sum of the numbers:
\[ 1002 + 1004 + 1006 + 1008 + 1010 \]

Calculating this:
\[ 1002 + 1004 = 2006 \]
\[ 2006 + 1006 = 3012 \]
\[ 3012 + 1008 = 4020 \]
\[ 4020 + 1010 = 5030 \]
So, the sum of the numbers is \( 5030 \).

### Step 2: Calculate the mean
Next, we calculate the mean using the formula:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
Here, the number of observations is 5:
\[ \text{Mean} = \frac{5030}{5} = 1006 \]

### Step 3: Set up the equation
According to the problem, the mean is also given by the equation:
\[ n^3 + 6 = 1006 \]

### Step 4: Solve for \( n^3 \)
Now, we can rearrange the equation to solve for \( n^3 \):
\[ n^3 = 1006 - 6 \]
\[ n^3 = 1000 \]

### Step 5: Solve for \( n \)
To find \( n \), we take the cube root of both sides:
\[ n = \sqrt[3]{1000} \]
Since \( 1000 = 10^3 \):
\[ n = 10 \]

### Final Answer
Thus, the value of \( n \) is \( 10 \).

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