If x > 1, y > 1, z > 1 are in G.P., then show that 1/(1 + log x), 1/(1 + log y), 1/(1 + log z) are in H.p.

Question

If x > 1, y > 1, z > 1 are in G.P., then show that \(\frac{1}{1+\text{log}\,x}\), \(\frac{1}{1+\text{log}\,y}\), \(\frac{1}{1+\text{log}\,z}\) are in H.p.

Answer 1

x, y, z are in G.P. 

⇒ y² = xz ⇒ 2 log y = log x + log z 

⇒ log x, log y, log z are in A.P. 

⇒ 1 + log x, 1 + log y, 1 + log z are in A.P.    (Adding 1 to each term)

 \(\frac{1}{1+\text{log}\,x}\), \(\frac{1}{1+\text{log}\,y}\), \(\frac{1}{1+\text{log}\,z}\) are in H.p.