The problem
The sides of a circular track contain a sequence of cans of gasoline. The total amount in the cans is sufficient to enable a certain car to make one complete circuit of the track, and it could all fit into the car's gas tank at one time. Use mathematical induction to prove that it is possible to find an initial location for placing the car so that it will be able to traverse the entire track by using the various amounts of gasoline in the cans that it encounters along the way.
My solution
Given any sequence of cans, we know that the sum of all the gasoline will traverse the car around the entire track. What is not known is the initial location of the car.
Basis step: The track has $1$ can, therefore the car will be able to traverse the track from some starting point.
$S(1) =$ some starting point; $S(1) = p_1$, where $p_1$ is the initial starting point.
Induction hypothesis: $S(k) = p_k$, such that $p_k$ is some starting point that, when traversing the track, if at any point on the track the car runs out of fuel, that point will not work. In order to find $p_k$, one will have to exhaust all possible finite points on the track in order to find the initial point that will traverse the car around the track where some nth tank will have enough fuel to get to the next tank.
Thus, $S(k + 1) = p_{k + 1}$, where $p_{k + 1}$ is some point. If at any point in traversing the track the car runs out of fuel, the chosen $p_{k + 1}$ will not work. A car will run out of fuel if it cannot travel from some $n$-th fuel tank to the $(n + 1)$-th fuel tank.
Done.
Is this right?