Question 170·Hard·Advanced Math

Pick $k=\frac12$ as a check value. In Desmos, type $k=\frac12$, then $n_1/(n_1-4)\sim k^2$. The $\sim$ tells Desmos to find a value for $n_1$; under PARAMETERS, it shows $n_1\approx-1.333$. So your formula should give a negative result at this $k$. One check cannot prove the formula, though.

The denominator $n-4$ cannot be zero, so $n\ne4$. Multiply both sides by that nonzero expression to clear the fraction:

Multiply both terms inside $n-4$ by $k^2$:

Subtract $k^2n$ from both sides to put the $n$-terms together:

Both terms on the left contain $n$. To factor means to write them as one product. Factor out $n$: $n(1-k^2)=-4k^2$.When the target appears in two terms, factor it out before dividing.

Multiply both sides by $-1$ so the factor beside $n$ becomes $k^2-1$:

If $k^2=1$, the given equation would require $n=n-4$, which is impossible. Otherwise, divide both sides by $k^2-1$: $n=\frac{4k^2}{k^2-1}$. This expresses $n$ in terms of $k$. Choice B.