The average of all the perfect squares upto 100 is:

Options

A. 38.5
B. 1000
C. 100
D. 385

Answer

To find the average of all the perfect squares up to 100, we can follow these steps:

### Step 1: Identify the perfect squares up to 100.
The perfect squares up to 100 are the squares of the integers from 1 to 10. These are:
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)
- \(7^2 = 49\)
- \(8^2 = 64\)
- \(9^2 = 81\)
- \(10^2 = 100\)

So, the perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

### Step 2: Add the perfect squares.
Now, we need to add these perfect squares together:
\[ 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 \]
Calculating this step-by-step:
- \(1 + 4 = 5\)
- \(5 + 9 = 14\)
- \(14 + 16 = 30\)
- \(30 + 25 = 55\)
- \(55 + 36 = 91\)
- \(91 + 49 = 140\)
- \(140 + 64 = 204\)
- \(204 + 81 = 285\)
- \(285 + 100 = 385\)

So, the sum of the perfect squares is \(385\).

### Step 3: Count the number of perfect squares.
We have identified that there are 10 perfect squares up to 100.

### Step 4: Calculate the average.
To find the average, we divide the sum of the perfect squares by the number of perfect squares:
\[ \text{Average} = \frac{\text{Sum of perfect squares}}{\text{Number of perfect squares}} = \frac{385}{10} = 38.5 \]

Thus, the average of all the perfect squares up to 100 is **38.5**.

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