The sum of all natural numbers from 75 to 97 is:

Options

A. 1598
B. 1978
C. 1798
D. 1958

Answer

To find the sum of all natural numbers from 75 to 97, we can use the formula for the sum of the first n natural numbers and apply it to our specific range. Here's a step-by-step solution:

### Step 1: Identify the range
We need to find the sum of natural numbers from 75 to 97, inclusive.

### Step 2: Use the formula for the sum of natural numbers
The formula for the sum of the first n natural numbers is:
\[ S_n = \frac{n(n + 1)}{2} \]
where \( S_n \) is the sum of the first n natural numbers.

### Step 3: Calculate the sum from 1 to 97
First, we calculate the sum of natural numbers from 1 to 97:
\[ S_{97} = \frac{97 \times (97 + 1)}{2} = \frac{97 \times 98}{2} = \frac{9506}{2} = 4753 \]

### Step 4: Calculate the sum from 1 to 74
Next, we calculate the sum of natural numbers from 1 to 74:
\[ S_{74} = \frac{74 \times (74 + 1)}{2} = \frac{74 \times 75}{2} = \frac{5550}{2} = 2775 \]

### Step 5: Find the sum from 75 to 97
Now, we find the sum of natural numbers from 75 to 97 by subtracting the sum from 1 to 74 from the sum from 1 to 97:
\[ \text{Sum from 75 to 97} = S_{97} - S_{74} = 4753 - 2775 = 1978 \]

### Final Answer
The sum of all natural numbers from 75 to 97 is **1978**.

---