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

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

3. If \(a, b, c, d\) are in continuous geometric progression, show that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\)

4. If \(a, b, c, d\) are in continuous geometric progression, show that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\)

5. If \(a, b, c, d\) are in continuous proportion, prove that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\).

6. If \(a, b, c, d\) are in continuous proportion, then show that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\).

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

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

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

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

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

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

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

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

15. If \(a, b, c\) are in continued geometric progression, prove that \(\cfrac{1}{b} = \cfrac{1}{b - a} + \cfrac{1}{b - c}\).

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

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

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