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

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

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

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

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

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

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

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 \(a : b = c : d\), then prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

12. If \(a, b, c, d\) are in continued proportion (i.e., in geometric progression), then the value of \(\cfrac{adc(a + b + c)}{ab + bc + ca}\) is —

(a) \(\cfrac{1}{a^2}\) (b) \(\cfrac{1}{c^2}\) (c) \(c^2\) (d) \(a^2\)

13. If \(a : b = c : d\), prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

14. If \(a:b=c:d\), then prove that \((a^2+c^2)(b^2+d^2)=(ab+cd)^2\).

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

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

17. If \(a : b = c : d\), prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

18. If \(a, b, c, d\) are in continued proportion, then what is the value of \(\frac{abc(a + b + c)}{ab + bc + ca}\)?

(a) \(\cfrac{1}{a^2}\) (b) \(\cfrac{1}{c^2}\) (c) \(b^2\) (d) \(a^2\)

19. If \(a, b, c, d\) are in continuous proportion, then what is the value of \(\cfrac{a^2 + b^2 + c^2}{b^2 + c^2 + d^2}\)?

(a) \(\cfrac{a}{c}\) (b) \(\cfrac{b}{c}\) (c) \(\cfrac{c}{d}\) (d) \(\cfrac{b}{d}\)

20. In triangle △ABC, ∠A is a right angle and BP and CQ are medians. Prove that: \[ 5BC^2 = 4(BP^2 + CQ^2) \]

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

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

23. Assume, \(\cfrac{a}{b} = \cfrac{b}{c} = \cfrac{c}{d} = k\) (where \(k \ne 0\)) So, \(a = bk,\; b = ck,\; c = dk\) \(\therefore b = dk \cdot k = dk^2\) and \(a = dk^2 \cdot k = dk^3\)

Left-hand side \[ = (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = \{(dk^3)^2 + (dk^2)^2 + (dk)^2\} \cdot \{(dk^2)^2 + (dk)^2 + d^2\} \] \[ = \{d^2k^6 + d^2k^4 + d^2k^2\} \cdot \{d^2k^4 + d^2k^2 + d^2\} = d^2k^2(k^4 + k^2 + 1) \cdot d^2(k^4 + k^2 + 1) \] \[ = d^4k^2(k^4 + k^2 + 1)^2 \] Right-hand side \[ = (ab + bc + cd)^2 = (dk^3 \cdot dk^2 + dk^2 \cdot dk + dk \cdot d)^2 = (d^2k^5 + d^2k^3 + d^2k)^2 = \{d^2k(k^4 + k^2 + 1)\}^2 = d^4k^2(k^4 + k^2 + 1)^2 \] ∴ Left-hand side = Right-hand side (Hence Proved)

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

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