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Question
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
Solution
Let r be the common ratio of the given G.P.
Then `b/a = c/b = d/c` = r
⇒ b = ar, c = br = ar2, d = cr = ar3
Now, a2 – b2 = a2 – a2r2
= a2(1 – r2)
b2 – c2 = a2r2 – a2r4
= a2r2 (1 – r2)
And c2 – d2 = a2r4 – a2r6
= a2r4(1 – r2)
Therefore, `(b^2 - c^2)/(a^2 - b^2) = (c^2 - d^2)/(b^2 - c^2)` = r2
Hence, a2 – b2, b2 – c2, c2 – d2 are in G.P.
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