The three cyclists are cycling from Burgersfort to Cape Town, they are planning to take their first rest after 100km, second rest after 150km and third rest after 200km. The pattern is as follows: 100, 150, 200

The distances between rests form an arithmetic progression: 100 km, 150 km, 200 km, ...
The first term ($a$) is 100 km.
The common difference ($d$) is $150 - 100 = 50$ km.
We assume each segment between rests represents a day's travel.

The total distance ($S_N$) covered after $N$ days is given by:

We have $S_N = 1750$ km, $a = 100$ km, and $d = 50$ km.

Rearrange into a standard quadratic equation:

Divide the entire equation by 25 to simplify:

Factor the quadratic equation:

This gives two possible solutions for $N$:

Since the number of days cannot be negative, we take the positive value.

The cyclists are expected to take $\boxed{\mathbf{7 \text{ days}}}$ for their trip.