​Directions: Choose the correct alternative.​

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

Correct option is D

Given:

We need to find the sum of all natural numbers from 75 to 97.

Formula used:

The sum of an arithmetic series (sequence of numbers with a constant difference between them) can be calculated using the formula:

​$$S_n = \frac{n}{2} \times (a + l)$$

$$l = a + (n - 1) \times d$$

where:

n is the number of terms,
a is the first term,
l is the last term.

Solution:

First term, a = 75,
Last term, l = 97,

The common difference between the terms is 1 (since the numbers are consecutive).

Put the values in formula ,

$$l = a + (n - 1) \times d$$​​

​$$97 = 75 + (n - 1) \times 1$$

97 − 75 = n − 1

n = 23

$$S_n = \frac{n}{2} \times (a + l)$$

$$S_n = \frac{23}{2} \times (75 + 97)$$​

​$$S_n= \frac{23}{2} \times 172$$​

​$$S_n = 23 \times 86$$​

​$$S_n = 1978​​​$$

Thus, the correct answer is (d).