Find the sum of all multiples of 7 lying between 300 and 700.

Answer

To find the sum of all multiples of 7 lying between 300 and 700, we can follow these steps:

### Step 1: Identify the first multiple of 7 greater than 300
To find the first multiple of 7 that is greater than 300, we can divide 300 by 7 and round up to the nearest whole number:
\[ \text{First multiple} = 7 \times \lceil \frac{300}{7} \rceil \]
Calculating \( \frac{300}{7} \approx 42.857 \), rounding up gives us 43.
\[ \text{First multiple} = 7 \times 43 = 301 \]

### Step 2: Identify the last multiple of 7 less than 700
Next, we find the largest multiple of 7 that is less than 700:
\[ \text{Last multiple} = 7 \times \lfloor \frac{700}{7} \rfloor \]
Calculating \( \frac{700}{7} = 100 \), we take the floor value:
\[ \text{Last multiple} = 7 \times 100 = 700 \]
Since we want the last multiple less than 700, we take \( 7 \times 99 \):
\[ \text{Last multiple} = 7 \times 99 = 693 \]

### Step 3: Determine the number of terms in the arithmetic progression (AP)
The multiples of 7 between 301 and 693 form an arithmetic progression where:
- First term \( a = 301 \)
- Last term \( l = 693 \)
- Common difference \( d = 7 \)

To find the number of terms \( n \), we use the formula for the nth term of an AP:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ 693 = 301 + (n-1) \times 7 \]
Solving for \( n \):
\[ 693 - 301 = (n-1) \times 7 \\ 392 = (n-1) \times 7 \\ n-1 = \frac{392}{7} = 56 \\ n = 56 + 1 = 57 \]

### Step 4: Calculate the sum of the arithmetic progression
The sum \( S_n \) of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (a + l) \]
Substituting the values we found:
\[ S_{57} = \frac{57}{2} \times (301 + 693) \\ = \frac{57}{2} \times 994 \\ = 57 \times 497 \\ = 28329 \]

### Final Answer
The sum of all multiples of 7 lying between 300 and 700 is **28329**.