Let p = ln(x), q = ln(x³) and r = ln(x⁵), where x > 1. Which of the following statements is/are correct? 

I. p, q and r are in AP.

Il. p, q and r can never be in GP.

Select the answer using the code given below.  

Concept:

Arithmetic Progression (AP):

• The terms are in AP if the difference between consecutive terms is constant.
• The common difference is given by: \(d = q - p = r - q\)

Geometric Progression (GP):

• The terms are in GP if the ratio between consecutive terms is constant.
• The common ratio is given by: \(r = q \times p\)

Calculation:

Given

p = ln(x),

q = ln(x3)  = 3 lnx

and r = ln(x5) = 5 lnx

Clearly, p - q  = r - q = \(2lnx\)

⇒ p, q, r are in AP

Also \(\frac{q}{p} ≠ \frac{r}{p} \because \frac{q}{p} = 3 , \frac{r}{p} = \frac{5}{3}\)

∴ p, q, r can never be in GP

∴ Option (c) is correct.