Find the average of cubes of first 7 consecutive natural numbers.

Options

A. 112
B. 110
C. 116
D. 113

Answer

To find the average of the cubes of the first 7 consecutive natural numbers, we can follow these steps:

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

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

### Step 3: Sum the cubes
Next, we will sum these cubes:
\[ 1 + 8 + 27 + 64 + 125 + 216 + 343 \]

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: Calculate the average
To find the average, we divide the total sum of the cubes by the number of terms (which is 7):
\[ \text{Average} = \frac{\text{Sum of cubes}}{\text{Number of terms}} = \frac{784}{7} \]

Calculating this gives:
\[ \frac{784}{7} = 112 \]

### Final Answer
The average of the cubes of the first 7 consecutive natural numbers is \(112\).

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