By Abraham P Hillman

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The method of mathematical induction is based on something that may be considered one of the axioms for the positive integers: If a set S contains 1, and if, whenever S contains an integer k, S contains the next integer k + 1, then S contains all the positive integers. It can be shown that this is equivalent to the principle that in every non-empty set of positive integers there is a least positive integer. Example 1. Find and prove by mathematical induction a formula for the sum of the first n cubes, that is, 13 + 23 + 33 + ...

For n = 4, these ratios are: (2) 4/1, 6/4 = 3/2, 4/6 = 2/3, 1/4. For n = 5, they are (3) 5/1, 10/5 = 2, 10/10 = 1, 5/10 = 1/2, 1/5. The ratios in (3) have the same pattern as those in (2) if they are rewritten as 5/1, 4/2, 3/3, 2/4, 1/5. It is easily seen that this pattern also holds on the line for n = 8, and that the coefficients on that line are therefore: 1, (4) The binomial coefficient 8 8 7 8 7 6 8 7 6 5 , @ , @ @ , @ @ @ , ... 1 1 2 1 2 3 1 2 3 4 8 8@7@6 ' can be rewritten as 3 1@2@3 8@7@6@5@4@3@2@1 8!

A 60-mile trip was made at 30 miles per hour and the return at 20 miles per hour. (a) How many hours did it take to travel the 120 mile round trip? (b) What was the average speed for the round trip? 30. Find x, given that 1/30, 1/x, and 1/20 are in arithmetic progression. What is the relation between x and the answer to Part (b) of problem 29? 31. Verify the factorization 1 - x7 = (1 - x)(1 + x + x2 + x3 + x4 + x5 + x6) and use it with x = 1/2 to find a compact expression for 1% 1 1 % 2 2 2 % 1 2 3 % 1 2 4 % 1 2 5 % 1 2 6 .