TMUA Official 2017 Paper 2 · Question 18

Consider the following problem:

Solve the inequality $\left(\frac14\right)^n<\left(\frac1{32}\right)^{10}$ , where $n$ is a positive integer.

A student produces the following argument:

\begin{gathered} \left(\frac14\right)^n<\left(\frac1{32}\right)^{10}\\[3pt] \underset{\text{(I)}}{\Downarrow}\\[3pt] \log_{\frac12}\left(\frac14\right)^n<\log_{\frac12}\left(\frac1{32}\right)^{10}\\[3pt] \underset{\text{(II)}}{\Downarrow}\\[3pt] n\log_{\frac12}\left(\frac14\right)<10\log_{\frac12}\left(\frac1{32}\right)\\[3pt] \underset{\text{(III)}}{\Downarrow}\\[3pt] n<\frac{10\log_{\frac12}\left(\frac1{32}\right)}{\log_{\frac12}\left(\frac14\right)}\\[3pt] \underset{\text{(IV)}}{\Downarrow}\\[3pt] n<\frac{10\times5}2=25\\[3pt] \underset{\text{(V)}}{\Downarrow}\\[3pt] 1\leq n\leq24 \end{gathered}

Which step (if any) in the argument is invalid?

AThere are no invalid steps; the argument is correct

BOnly step (I) is invalid; the rest are correct

COnly step (II) is invalid; the rest are correct

DOnly step (III) is invalid; the rest are correct

EOnly step (IV) is invalid; the rest are correct

FOnly step (V) is invalid; the rest are correct

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