If x, y, z are in G.P. and (log x – log 2y), (log 2y – log 3z) and (log 3z – log x) are in A.P., -Maths 9th

If x, y, z are in G.P. and (log x – log 2y), (log 2y – log 3z) and (log 3z – log x) are in A.P., -Maths 9th
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(d) obtuse angledx, y, z are in G.P. ⇒ y2 = xz               ...(i) (log x – log 2y), (log 2y – log 3z) and (log 3z – log x) are in A.P. ⇒ 2(log 2y – log 3z) = (log x – log 2y) + (log 3z – log x) ⇒ 3 log 2y = 3 log 3z ⇒ log 2y = log 3z ⇒ y = \(rac{3}{2}\)z ∴ Putting the value of y in (i), we have\(\big(rac{3}{2}z\big)^2\) = xz ⇒ x = \(rac{9}{4}\)z.Now, by the cosine rule of triangles,Cos A = \(rac{y^2+z^2-x^2}{2yz}\), where x is the length of the side opposite ∠A.∵ cos A is less than 0, i.e, negative, ∠A is obtused and the triangle is obtuse angled.
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