Calculate the average of the cubes of first 9 natural numbers .

Answer

To calculate the average of the cubes of the first 9 natural numbers, we will follow these steps:

### Step 1: Identify the first 9 natural numbers
The first 9 natural numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9.

### Step 2: Calculate the cubes of these numbers
Now we will calculate the cube of each of these numbers:
- \(1^3 = 1\)
- \(2^3 = 8\)
- \(3^3 = 27\)
- \(4^3 = 64\)
- \(5^3 = 125\)
- \(6^3 = 216\)
- \(7^3 = 343\)
- \(8^3 = 512\)
- \(9^3 = 729\)

### Step 3: Sum the cubes
Next, we will sum all the cubes calculated in the previous step:
\[ 1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 \]

Calculating the sum:
- \(1 + 8 = 9\)
- \(9 + 27 = 36\)
- \(36 + 64 = 100\)
- \(100 + 125 = 225\)
- \(225 + 216 = 441\)
- \(441 + 343 = 784\)
- \(784 + 512 = 1296\)
- \(1296 + 729 = 2025\)

So, the sum of the cubes is \(2025\).

### Step 4: Calculate the average
To find the average, we divide the sum of the cubes by the number of natural numbers (which is 9):
\[ \text{Average} = \frac{\text{Sum of cubes}}{\text{Number of terms}} = \frac{2025}{9} \]

Calculating the average:
\[ \frac{2025}{9} = 225 \]

### Final Answer
The average of the cubes of the first 9 natural numbers is **225**.

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