8 Inductive Charging Lemma There are n cars on a circular track. Among all of them, they have exactly enough fuel (in total) for one car to circle the track. Prove, using whatever method you want, that there exists at least one car that has enough fuel to reach the next car along the track.

There is a car that can complete a lap by obtaining gas from the other cars as it travels around the track, when there are n cars in the group" by proving that the statement is true when n = 1 and proving that the statement is true for n = k + 1 when given that the statement is true for n = k.

Step-by-step explanation:

Given:

There is enough fuel for one car to complete a lap among a group of cars.  

To proof:

There is a car that can complete a lap by obtaining gas from the other cars as it travels around the track.  

PROOF  BY INDUCTION

Let P(n) be the statement "There is a car that can complete a lap by obtaining gas from the other cars as it travels around the track, when there are n cars in the group".  

Basis step n = 1  

If there is one car in the group, then this one car needs to have enough fuel to complete the lap.

Thus P(I) is true.

Inductive step Let P(k) be true.

There is a car that can complete a lap by obtaining gas from the other cars as it travels around the track, when there are k cars in the group  

We need to proof that P(k + 1) is true.

When there are k+ 1 cars in the group, then there needs to be at least one car x that has enough gas to drive to the next car y on the track (as there won't be enough fuel among the group of cars to complete the track if this is not true).  

Let us then add the fuel of car y to the fuel of car x. Then there are k remaining cars in the group remaining, while there is a car z that can complete a lap by obtaining gas from the other cars (as P(k) is true).  

Moreover, this car z is then also the car that can complete a lap by obtaining gas from the other cars among the group of k + 1 cars.  

Thus P(k + 1) is true.  

Conclusion:

By the principle of mathematical induction, P(n) is true for all positive integers n. 0