A car is moving at 40 km/h. A fly at 100 km/h, starts from wall towards the car(20 km away)flies to car and back. How many trips can it make?

It has made infinitely many trips. Every trip will be shorter than the last, but the fly will always reach the wall before the car, so it will always have room for one more trip. And one more. And one more.

The total length the fly flies is 50km, as the car crashes into the wall exactly 30 minutes after the whole experiment started.

Suppose the fly is at the wall. And suppose this will be the last trip or partial trip of the fly. Suppose the car is $h$ miles away.

The fly and and car have a combined speed of $140 \frac {km}{hr}$ so the fly reaches the car in $\frac h{140}$ hours. In that time the car has traveled $40\frac h{140} = \frac 27h$ and is now $h-\frac 27h = \frac 57h$ from the wall. So the fly heads back to the wall.

As the trip back is just as far this takes $\frac h{140}$ hours and the car has traveled another $\frac 27h$ and is now $\frac 37h$ from the wall. [1]

So the fly starts another trip, contradicting that this was his last. So the fly never makes a last trip an instead there are an infinite number of trips.

Figuring out how far the fly flies is a matter of noting the car is on a straight path and travels $20km$ at $40 kmh$ so this takes $30$ minutes. The fly no matter how many times (infinitely many) it zigs will travel at $100kmh$. So in $30$ minutes it flies $50 km$.

If one wishes to set this up as an infinite sum.....

Each trip the fly flies $\frac {10}7$ of the distance the car was away. And each trip the car is $\frac {3}{7}$ of the distance it was before. So the distance the fly travels is $\sum_{k=0}^{\infty} \frac {10}7*(\frac {3}{7})^k*20$ which if I did this correctly is

$\frac {10}7*20(\sum_{k=0}^{\infty} (\frac {3}{7})^k)= \frac {200}{7}\frac 1{1-\frac {3}{7}} = \frac {200}{7}\frac {7}{4}= 50$km.

[1](for the record, in this time, $\frac 1{70}h$ hours, the car has traveled $\frac 47h$ and the fly has travelled $\frac {10}7h$).

Simple way to calculate it: It takes the car 30 minutes to travel the 20km to the wall at 40km/hr. The fly, traveling at 100km/hr will travel 50km in those same 30 minutes.

Edit: There will be an infinite number of trips. It's similar to how Zeno's paradox works where the trips get shorter and shorter and eventually take an infinitely small amount of time. But all those infinitely small trips end up as a finite amount of distance.