1Define geometric mean with an example.
2Determine a geometric mean between p and q.
3Find a geometric mean between \( \frac{1}{10} \) and 10.
4Find the 10th term of a GP whose 9th term is \( \frac{1}{5} \) and 11th term is 125.
5The geometric mean between \( \frac{1}{9} \) and b is 2. Find the value of b.
6There are 3 GMs in a GP in which the first term is 10 and the common ratio is \( \frac{1}{2} \). Find the first mean only.
7Find the AM and GM between 4 and 16. Also write their difference.
8The arithmetic mean of a and 24 is 15. Find the geometric mean (GM).
9Insert 2 GMs between 6 and 48.
10Insert 3 GMs between 5 and 80.
11Find the values of x and y when \( \frac{1}{2}, x, 2, y \) are in GP.
12Insert 7 GMs between \( \frac{1}{16} \) and 16 and find the difference between the 5th mean and the first mean.
13Insert 6 GMs between \( \frac{1}{2} \) and 64 and compare the 1st mean and the 6th mean.
14There are 4 geometric means between 4 and q. If the 2nd mean is 36, find the value of q and the remaining means.
15Some geometric means are inserted between 5 and 80. Find the number of means if the third mean is 40. Also find the remaining means.
16The AM between two positive numbers is 50 and the GM is 40. Find the numbers.
17The AM between two natural numbers is 45 and the GM is 27. Find the numbers.
18There are n geometric means between \( \frac{1}{81} \) and 81. If the ratio of the 3rd mean to the last mean is 1 : 81, find the value of n.
19There are n geometric means between \( \frac{27}{16} \) and \( \frac{8}{81} \). If the ratio of the (n - 1)th mean to the 2nd mean is 8 : 27, find the value of n.
20Prove that the arithmetic mean of two positive numbers is never less than their geometric mean. When are they equal?
21Why is there no real geometric mean between -4 and 9?
22Show with the areas of a rectangle and a square that the geometric mean of 4 and 9 is 6.
23If 4 geometric means are inserted between a and b, how many terms does the GP have, and what power appears in the formula for r?