Find all possible ways of expressing 100 as the sum of consecutive natural numbers.

Let the consecutive natural numbers be:

$$a, a + 1, a + 2, \dots, a + (n - 1)$$

where, $a =$ first term and $n =$ number of terms Since these are in AP:

$$S_n = \frac{n}{2}[2a + (n - 1)]$$

Given sum = 100,

$$\frac{n}{2}[2a + n - 1] = 100$$

$$n(2a + n - 1) = 200$$

Now check factors of 200 so that $a$ becomes a natural number.

Case 1: $n = 5$

$$5(2a + 4) = 200$$

$$2a + 4 = 40$$

$$2a = 36$$

$$a = 18$$

So, $100 = 18 + 19 + 20 + 21 + 22$

Case 2: $n = 8$

$$8(2a + 7) = 200$$

$$2a + 7 = 25$$

$$2a = 18$$

$$a = 9$$

So, $100 = 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16$

Hence, the possible ways are:

$$100 = 18 + 19 + 20 + 21 + 22$$

and $100 = 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16$