1. If \(a, b, c\) are in continued geometric progression (i.e., in successive proportion), then prove that \[ \cfrac{1}{b^2} = \cfrac{1}{b^2 - a^2} + \cfrac{1}{b^2 - c^2} \]

2. If \(a, b, c\) are in continued geometric progression, prove that \[ a^2b^2c^2\left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

3. If \(a, b, c\) are in continued geometric progression, then prove that \[ a^2 b^2 c^2 \left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

4. If \(a, b, c, d\) are in continued geometric progression, prove that \[(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2\]

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

6. If a rectangular parallelepiped has volume \(v\), and its length, width, and height are \(a, b, c\) respectively, with a total surface area \(s\), prove that: \(\cfrac{1}{v}=\cfrac{2}{3}\left(\cfrac{1}{a}+\cfrac{1}{b}+\cfrac{1}{c}\right)\).

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

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

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

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

11. If \(a, b, c, d\) are consecutive terms in a geometric progression, prove that \((a^2 - b^2)(c^2 - b^2) = (b^2 - c^2)^2\).

12. If \( a, b, c \) are in continuous proportion, prove that \( a^2b^2c^2\left(\cfrac{1}{a^3}+\cfrac{1}{b^3}+\cfrac{1}{c^3}\right)=a^3+b^3+c^3 \).

13. If \(a, b, c, d\) are consecutive terms in geometric progression, prove that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\).

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 \(a, b, c, d\) are in continued proportion, then prove that \[ (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2 \]

16. If \(a, b, c, d\) are in continued geometric progression (i.e. in sequence with common ratio), then prove that \[ (a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2 \]