1. \(x^3y, x^2y^2,\) and \(xy^3\) are in continued proportion.

2. If 7, x, y, 189 are in continued proportion, then the values of x and y respectively will be:

(a) 63,21 (b) 21,23 (c) 21,63 (d) 23,21

3. Four quantities are in continuous proportion. If the first and second quantities are 3 and 2 respectively, then the fourth quantity will be:

(a) \(\cfrac{2}{3}\) (b) \(\cfrac{8}{9}\) (c) \(\cfrac{4}{3}\) (d) \(\cfrac{10}{12}\)

4. If \(x, 2x, 3\), and \(y\) are in continued proportion, then what will be the value of \(y\)?

(a) \(4x\) (b) \(6x\) (c) 4 (d) 6

5. Which number should be added to each of 4, 6, and 10 so that the resulting sums are in continued proportion?

6. If 8, 14, \(7x + 2\), and 28 are in continued proportion, then what is the value of \(x\)?

(a) 1 (b) 3 (c) 2 (d) 4

7. If \(4\), \(5 + x\), \(2x + 1\), and \(14\) are in continued proportion and \(x\) is a positive number, then what is the value of \(x\)?

(a) 1 (b) 2 (c) 4 (d) 3

8. From each of the numbers 6, 10, 9, and 16, if the same number is subtracted, the resulting values are in proportion. What is that number?

(a) 1 (b) 2 (c) 3 (d) 4

9. If the numbers \((2 + x)\), \((8 + x)\), and \((26 + x)\) are in continued proportion, then what is the value of \(x\)?

(a) 4 (b) 3 (c) 2 (d) 1

10. If \(x\), 12, \(y\), and 27 are in continued proportion, find the negative values of \(x\) and \(y\).

11. If \(a, b, c,\) and \(d\) are in continued geometric proportion, then prove that \[ (b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2 \]

12. \(p^3q, p^2q^2\), and \(p q^3\) are in continuous proportion.

13. If \(a, b, c,\) and \(d\) are in continued proportion, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\).

14. If \(a, b, c\), and \(d\) are in continued proportion, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\)

15. If \(a, b, c,\) and \(d\) are in continued geometric progression, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\)

16. Determine the fifth number if the first two of five continued proportional numbers are 2 and 6.

17. Write with calculation how much should be added to each of 6, 15, 20, and 43 so that the resulting sums are in proportion.

18. Determine how much must be subtracted from each of 23, 30, 57, and 78 so that the resulting differences are in proportion.

19. Determine how much must be subtracted from each of \(p, q, r,\) and \(s\) so that the resulting differences are in proportion.

20. If \(x, 12, y, 27\) are in continued proportion, find the positive values of \(x\) and \(y\).

21. If \(x ∝ \cfrac{1}{y}\) and \(y ∝ \cfrac{1}{z}\), then determine whether \(x\) and \(z\) are in direct or inverse proportion.

22. If three real numbers a, b, and c are in continued geometric progression, then the value of the mean proportional b will be undefined when a and c have opposite signs.

23. If \(ad = bc\), show that \((a - b) : b\) and \((c - d) : d\) are in proportion.

24. If \(x, y\) are in direct proportion, and \(y, z\) are in inverse proportion, then \(x, z\) will be in inverse proportion.