Question

What is the sum for all multiples of 5 between 300 and 750

Answer 1

To find the sum of all multiples of 5 between 300 and 750, we first identify the smallest and largest multiples of 5 within that range.

The smallest multiple of 5 greater than or equal to 300 is:
\[ 300 \div 5 = 60 \quad \text{(which is already an integer)} \]
Thus, the smallest multiple of 5 is \( 300 \).

The largest multiple of 5 less than or equal to 750 is:
\[ 750 \div 5 = 150 \quad \text{(which is already an integer)} \]
Thus, the largest multiple of 5 is \( 750 \).

Now we need to find all multiples of 5 from 300 to 750. These multiples can be expressed as:
\[ 300, 305, 310, \ldots, 750 \]

This is an arithmetic series where:
- The first term \( a = 300 \)
- The last term \( l = 750 \)
- The common difference \( d = 5 \)

To find the number of terms \( n \) in the series, we use the formula for the \( n \)-th term of an arithmetic sequence:
\[ l = a + (n-1)d \]

Substituting the known values:
\[ 750 = 300 + (n-1) \cdot 5 \]
\[ 750 - 300 = (n-1) \cdot 5 \]
\[ 450 = (n-1) \cdot 5 \]
\[ n-1 = \frac{450}{5} = 90 \]
\[ n = 90 + 1 = 91 \]

Now that we know there are 91 terms, we can calculate the sum \( S_n \) of an arithmetic series, which is given by:
\[ S_n = \frac{n}{2} \cdot (a + l) \]

Substituting our values:
\[ S_{91} = \frac{91}{2} \cdot (300 + 750) \]
\[ S_{91} = \frac{91}{2} \cdot 1050 \]
\[ S_{91} = 91 \cdot 525 \]
\[ S_{91} = 47775 \]

Thus, the sum of all multiples of 5 between 300 and 750 is:
\[
\boxed{47775}
\]