A boy draws n squares with sides 1,2,3,4,5,…. In inches. The average area covered by these n square will be :

Options

A. a. `((n+1)/2)`
B. b. `((n+1)/2)((2n+1)/(3))`
C. c. `((n+1)/(2))((2n + 1)/(3))^(-1)`
D. d. `((n+1)/2) - 1((2n +1)/(3))`

Answer

To find the average area covered by the squares with sides 1, 2, 3, ..., n inches, we can follow these steps:

### Step 1: Understand the area of each square
The area \( A \) of a square with side length \( s \) is given by the formula:
\[ A = s^2 \]
So, for squares with sides 1, 2, 3, ..., n, the areas will be:
- Area of square with side 1: \( 1^2 = 1 \)
- Area of square with side 2: \( 2^2 = 4 \)
- Area of square with side 3: \( 3^2 = 9 \)
- ...
- Area of square with side n: \( n^2 \)

### Step 2: Calculate the total area of all squares
The total area \( T \) of the squares is the sum of the areas of all squares from 1 to n:
\[ T = 1^2 + 2^2 + 3^2 + ... + n^2 \]

### Step 3: Use the formula for the sum of squares
The formula for the sum of the squares of the first \( n \) natural numbers is:
\[ \text{Sum of squares} = \frac{n(n + 1)(2n + 1)}{6} \]
So, we can express the total area \( T \) as:
\[ T = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 4: Calculate the average area
The average area \( A_{\text{avg}} \) is given by the total area divided by the number of squares, which is \( n \):
\[ A_{\text{avg}} = \frac{T}{n} = \frac{\frac{n(n + 1)(2n + 1)}{6}}{n} \]

### Step 5: Simplify the expression
When we simplify the expression, we can cancel \( n \):
\[ A_{\text{avg}} = \frac{(n + 1)(2n + 1)}{6} \]

### Final Result
Thus, the average area covered by the squares is:
\[ A_{\text{avg}} = \frac{(n + 1)(2n + 1)}{6} \]

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