Q14 (d)

Criteria	Marks
Provides correct solution	3
Assumes $\log_n(n+1)$ is rational and attempts to eliminate the logarithm, or equivalent merit	2
Attempts to use proof by contradiction, or equivalent merit	1

Sample answer:

Suppose $\log_n(n+1)$ is rational.

Then

$$\log_n(n+1)=\frac pq,$$

where $p,q$ are integers and $q\ne0$.

$$\therefore n+1=n^{p/q}$$

so

$$(n+1)^q=n^p$$

If $n$ is even then RHS is even, LHS is odd—a contradiction.

If $n$ is odd then RHS is odd, LHS is even—a contradiction.

Hence in either case we have a contradiction, so $\log_n(n+1)$ is irrational.