Question

Suppose that among a group of cars on a circular track there is enough fuel for one car to complete a lap. Use mathematical induction to show that there is a car in the group that can complete a lap by obtaining gas from other cars as it travels around the track.

Short answer

By mathematical induction, the result \(P\left( n \right)\) is true for all positive integers \(n\).

Step-by-step solution

To recall the concepts and principles

Mathematical Induction:

The mathematical induction is defined as follows:

Step 1 (Base step): In this step, to prove that the statement is true for n=1.

Step 2(Inductive step): In this case, if the statement is true for n^th iteration, then to prove it is also true for (n+1)^st iteration.

It has given that among a group of cars on a circular track, there is enough fuel for one car to complete a lap.

To prove the result using principle of mathematical induction

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

Then by principle of mathematical induction,

For \(n = 1\):

Clearly, for one car in the group, there is enough fuel to complete a lap.

Thus, the result is true for \(n = 1\).

Hence, \(P\left( 1 \right)\) is true.

Now, consider the result is true for \(n = k\).

It means, there is a car in the group that can complete a lap by obtaining gas from other cars as it travels around the track, when there are \(k\)cars in the group.

Thus, \(P\left( k \right)\) is true.

Let us prove the result for \(n = k + 1\).

Let there be \(\left( {k + 1} \right)\) cars in the group.

Then there needs to be at least one car A which has enough gas to drive to the next car B on the track since there is enough fuel only for one car to complete a lap.

Since \(P\left( k \right)\) is true, let us add the fuel of car B to the fuel of car A.

Then there are \(k\) remaining cars in the group, while there is car C in the group that can complete a lap by obtaining gas from the other cars.

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

Thus, the result is true for \(n = k + 1\).

Hence, \(P\left( {k + 1} \right)\) is true.

Hence, by the principle of mathematical induction, the result \(P\left( n \right)\) is true for all positive integers \(n\).