If p,q and r (`pneq`) are terms ( not necessarily consecutive) of an A.P., then prove that there exists a rational number k such that `(r-q)/(q-p)`=k. hence, prove that the numbers `sqrt2,sqrt3 and sqrt5` cannot be the terms of a single A.P. with non-zero common difference.

Explanation

To prove that if \( p, q, r \) (where \( p \neq q \)) are terms of an arithmetic progression (A.P.), then there exists a rational number \( k \) such that

\[ \frac{r - q}{q - p} = k, \]

and to show that the numbers \( \sqrt{2}, \sqrt{3}, \sqrt{5} \) cannot be terms of a single A.P. with a non-zero common difference, we can follow these steps:

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