Find the sum of all positive multiples of 3, less than 50

Options

A. 400
B. 404
C. 408
D. 412

Answer

To find the sum of all positive multiples of 3 that are less than 50, we can follow these steps:

### Step 1: Identify the multiples of 3 less than 50
The positive multiples of 3 are:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48.

### Step 2: Count the number of terms
To find how many terms there are, we can use the formula for the nth term of an arithmetic sequence:
- The first term \( a = 3 \)
- The common difference \( d = 3 \)
- The last term \( l = 48 \)

We can find the number of terms \( n \) using the formula for the nth term:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 48 = 3 + (n - 1) \cdot 3 \]
\[ 48 - 3 = (n - 1) \cdot 3 \]
\[ 45 = (n - 1) \cdot 3 \]
\[ n - 1 = 15 \]
\[ n = 16 \]

### Step 3: Use the formula for the sum of an arithmetic series
The sum \( S_n \) of the first \( n \) terms of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we have:
\[ S_{16} = \frac{16}{2} \cdot (3 + 48) \]
\[ S_{16} = 8 \cdot 51 \]
\[ S_{16} = 408 \]

### Final Answer
The sum of all positive multiples of 3 less than 50 is **408**.

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