What is the value of `3^(2)+6^(2)+9^(2)+.....+60^(2)?`

Options

A. 25380
B. 25430
C. 25830
D. 25910

Answer

To find the value of \(3^2 + 6^2 + 9^2 + \ldots + 60^2\), we can follow these steps:

### Step 1: Identify the series
The series consists of the squares of multiples of 3 from \(3\) to \(60\). We can express the terms as:
\[ 3^2 + (2 \cdot 3)^2 + (3 \cdot 3)^2 + \ldots + (20 \cdot 3)^2 \]
This can be rewritten as:
\[ 3^2(1^2 + 2^2 + 3^2 + \ldots + 20^2) \]

### Step 2: Factor out \(3^2\)
Factoring out \(3^2\) gives us:
\[ S = 3^2 \left(1^2 + 2^2 + 3^2 + \ldots + 20^2\right) \]
Calculating \(3^2\):
\[ 3^2 = 9 \]
Thus, we have:
\[ S = 9 \left(1^2 + 2^2 + 3^2 + \ldots + 20^2\right) \]

### 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:
\[ \text{Sum} = \frac{n(n + 1)(2n + 1)}{6} \]
For \(n = 20\):
\[ \text{Sum} = \frac{20(20 + 1)(2 \cdot 20 + 1)}{6} \]
Calculating this step by step:
- \(20 + 1 = 21\)
- \(2 \cdot 20 + 1 = 41\)
- Now substitute into the formula:
\[ \text{Sum} = \frac{20 \cdot 21 \cdot 41}{6} \]

### Step 4: Simplify the expression
Now we simplify:
\[ \text{Sum} = \frac{20 \cdot 21 \cdot 41}{6} \]
First, simplify \(20\) and \(6\):
\[ \frac{20}{6} = \frac{10}{3} \]
Thus, we have:
\[ \text{Sum} = \frac{10 \cdot 21 \cdot 41}{3} \]

### Step 5: Calculate \(10 \cdot 21 \cdot 41\)
Calculating \(10 \cdot 21 = 210\):
\[ \text{Sum} = \frac{210 \cdot 41}{3} \]
Calculating \(210 \cdot 41\):
\[ 210 \cdot 41 = 8610 \]
Now divide by \(3\):
\[ \text{Sum} = \frac{8610}{3} = 2870 \]

### Step 6: Multiply by \(9\)
Now we return to our expression for \(S\):
\[ S = 9 \cdot 2870 \]
Calculating this gives:
\[ S = 25830 \]

### Final Answer
Thus, the value of \(3^2 + 6^2 + 9^2 + \ldots + 60^2\) is:
\[ \boxed{25830} \]