Given `1+2+3+4+…..+10=55` then the sum `6+12+18+24+…+60` is equal to

Options

A. 300
B. 655
C. 330
D. 455

Answer

To find the sum of the series \(6 + 12 + 18 + 24 + \ldots + 60\), we can follow these steps:

### Step 1: Identify the series
The series given is \(6 + 12 + 18 + 24 + \ldots + 60\). This series consists of multiples of 6.

### Step 2: Factor out the common term
We can factor out 6 from each term in the series:
\[ 6 + 12 + 18 + 24 + \ldots + 60 = 6(1 + 2 + 3 + 4 + \ldots + 10) \]
Here, we see that the series inside the parentheses is the sum of the first 10 natural numbers.

### Step 3: Use the formula for the sum of the first \(n\) natural numbers
The formula for the sum of the first \(n\) natural numbers is:
\[ S_n = \frac{n(n + 1)}{2} \]
For \(n = 10\):
\[ S_{10} = \frac{10(10 + 1)}{2} = \frac{10 \times 11}{2} = 55 \]

### Step 4: Substitute back into the equation
Now, substituting \(S_{10} = 55\) back into our factored equation:
\[ 6(1 + 2 + 3 + 4 + \ldots + 10) = 6 \times 55 \]

### Step 5: Calculate the final result
Now, we calculate:
\[ 6 \times 55 = 330 \]

### Conclusion
Thus, the sum \(6 + 12 + 18 + 24 + \ldots + 60\) is equal to \(330\).