Optimal jumping patterns

We consider the problem of finding optimal “jumping” patterns from 1 to N where there is a cost associated with each jump. This will be done for two cost functions, in the first case the cost of jumping from a to b will be (1 − q)/a for 0 < q < 1, while the second cost function will be b/a. For the first cost function we will show that all the jump lengths, except possibly the last jump, are between √ 2 and 19/4. This will imply that the number of jumps in this case is of order Θ ( min(lnN,− ln ln 1q ) ) . For the second cost function we will give some basic properties including bounds for the total cost of jumping from 1 to N .