The mean of `1^3. 2^3,3^3,4^3,5^3,6^3,7^3` is

Options

A. 20
B. 112
C. 56
D. 28

Answer

To find the mean of the cubes of the first seven natural numbers (1³, 2³, 3³, 4³, 5³, 6³, 7³), we can follow these steps:

### Step 1: Understand the formula for mean
The mean (average) is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]

### Step 2: Identify the values
We need to calculate the cubes of the first seven natural numbers:
- \(1^3 = 1\)
- \(2^3 = 8\)
- \(3^3 = 27\)
- \(4^3 = 64\)
- \(5^3 = 125\)
- \(6^3 = 216\)
- \(7^3 = 343\)

### Step 3: Calculate the sum of cubes
Now, we sum these values:
\[ 1 + 8 + 27 + 64 + 125 + 216 + 343 \]

Calculating this step-by-step:
- \(1 + 8 = 9\)
- \(9 + 27 = 36\)
- \(36 + 64 = 100\)
- \(100 + 125 = 225\)
- \(225 + 216 = 441\)
- \(441 + 343 = 784\)

So, the total sum of the cubes is:
\[ 784 \]

### Step 4: Count the number of values
There are 7 values (1 through 7).

### Step 5: Calculate the mean
Now, we can calculate the mean:
\[ \text{Mean} = \frac{784}{7} \]

Calculating this gives:
\[ \text{Mean} = 112 \]

### Final Answer
The mean of \(1^3, 2^3, 3^3, 4^3, 5^3, 6^3, 7^3\) is \(112\).

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