21. Find the geometric mean of each of the following sets of positive numbers:

(a) 6, 18.

(b) 2, 6, 18, 54.

(c) 2, 4, 8.

(d) 1, 2, 4, 8, 16.

22. Find the geometric mean of each of the following sets of numbers:

(a) 3, 4, 5.

(b) 3, 4, 5, 6.

(c) 1, 7, 7², 7³.

(d) a, ar, ar², ar³, ar⁴.

23. Find the geometric mean of 8, 27, and 125.

24. Find the geometric mean of a⁴, b⁴, c⁴, and d⁴.

25. Let b be the middle term of a geometric progression with 2m + 1 positive terms and let r be the common ratio. Show that:

(a) The terms are br^-m, br^-m+1, ... , br^-1, b, br, ... , br^m.

(b) The geometric mean of the 2m + 1 numbers is the middle term.

26. Show that the geometric mean of the terms in a geometric progression of positive numbers is equal to the geometric mean of any two terms equally spaced from the two ends of the progression.

27. Find a compact expression for the sum xⁿ + x^n-1y + x^n-2y² + ... + xy^n-1 + yⁿ.

28. Find a compact expression for the arithmetic mean of xⁿ, x^n-1y, x^n-2y², ..., xy^n-1, yⁿ.

29. 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?
 

Sunday, March 15, 1998