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$n$ cars are travelling down a narrow one-way street. We know that:

  • The distance $d$ between each two cars is the same.
  • The safe breaking distance $b$ is the minimum distance between two cars that is needed for the second car to stop on time if the car in front suddenly breaks.
  • $d < b$

Prove by induction or refute: if the first car suddenly stops moving, all cars will stop moving. Before you do the induction state the property $P$ you are using in the induction axiom.

What I came up so far:

Proof:

Base case: $n=1$

if there is only one car traveling down a narrow one way street, than obviously only 1 car will stop moving -> therefore the assertion is true

Induction step: $n= k+1$

I am stuck here---- need a hint

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It's unknown what will happen in the case $n\gt1$. As the cars don't have sufficient distance to break safely, the second car may crash into the first car. The given information doesn't determine what happens in this case; for instance, the brakes (not breaks) of both cars might break and the cars might skid along the (possibly inclined) street indefinitely.

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