If a, b, c, d are in geometric progression (GP), then prove that (bc)(bd) = (ca)(cd).

Proof:

Definition of a Geometric Progression (GP):
A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Given:
Let a, b, c, and d be in geometric progression, such that the common ratio is denoted by r. Therefore, we can write:

b = ar
c = ar^2
d = ar^3

Proof:
We need to prove that (bc)(bd) = (ca)(cd).

Step 1:
Calculate (bc)(bd):
(bc)(bd) = (ar)(ar^2)(ar)(ar^3)
= a^2r^3 * a^2r^4
= a^4r^7

Step 2:
Calculate (ca)(cd):
(ca)(cd) = (ar^2)(ar^3)(ar^2)(ar^3)
= a^2r^5 * a^2r^6
= a^4r^11

Step 3:
Compare the results from Step 1 and Step 2:
We have a^4r^7 = a^4r^11.

Step 4:
Simplify the equation:
r^7 = r^11

Step 5:
Divide both sides of the equation by r^7 (since r ≠ 0):
1 = r^4

Conclusion:
Since we have shown that 1 = r^4, we can conclude that the given equation (bc)(bd) = (ca)(cd) holds true when a, b, c, and d are in geometric progression.

Summary:
- Using the given sequence of numbers in geometric progression, we proved that (bc)(bd) = (ca)(cd).
- This proof involved substituting the values of b, c, and d in terms of a and r, the common ratio of the geometric progression.
- By simplifying the expression and comparing the results, we derived the equation r^7 = r^11.
- Dividing both sides by r^7, we obtained 1 = r^4, which confirms the initial statement.
- Therefore, the equation (bc)(bd) = (ca)(cd) is true for any geometric progression with a common ratio.

use four nos in GP concept a/r3 a/r ar ar3