Yes, your reasoning is logical and makes sense, but the question must have a typo because those IE values are the exact match for boron B, an element from period 2 right above aluminium Al (data from [1, p. 10-204], converted from $\pu{eV}$ to $\pu{kJ mol-1}$):
\begin{array}{lccccc}
\hline
\text{Element} & \mathrm{IE_1} & \mathrm{IE_2} & \mathrm{IE_3} & \mathrm{IE_4} & \mathrm{IE_5} \\
\hline
\ce{B} & 801 & 2427 & 3659 & 25022 & 32822 \\
\ce{Al} & 578 & 1816 & 2745 & 11580 & 14842 \\
\ce{Ga} & 580 & 1980 & 2965 & 6102 & 8299 \\
\ce{In} & 559 & 1821 & 2704 & 5210 & \\
\ce{Tl} & 589 & 1971 & 2878 & & \\
\hline
\end{array}

Another hint that it's not aluminium is the absolute value of the jump in energy between $\mathrm{IE_3}$ and $\mathrm{IE_4}$, which is too high for latter group 13 elements.
Probably the author meant group 13, not period 3, but that's just a speculation at this point.
References
- Haynes, W. M.; Lide, D. R.; Bruno, T. J. CRC Handbook of Chemistry and Physics: A Ready-Reference Book of Chemical and Physical Data.; 2017; Vol. 97. ISBN 978-1-4987-5429-3.
Appendix
Source code for the plot, pgfplots:
\documentclass{standalone}
\usepackage{pgfplots}
\usepackage{siunitx}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
title={Successive ionization energies of group 13 elements},
xmin=1, xmax=5,
ymin=0, ymax=35000,
xlabel = $i$,
ylabel = {$\mathrm{IE}_i$/\si{\kilo\joule\per\mole}},
minor tick num = 1,
mark = *,
grid,
legend cell align = {left},
legend pos = north west,
]
\addplot coordinates {
(1,801)
(2,2427)
(3,3659)
(4,25022)
(5,32822)
};
\addlegendentry{Boron}
\addplot coordinates {
(1,578)
(2,1816)
(3,2745)
(4,11580)
(5,14842)
};
\addlegendentry{Aluminium}
\addplot coordinates {
(1,580)
(2,1980)
(3,2965)
(4,6102)
(5,8299)
};
\addlegendentry{Gallium}
\addplot coordinates {
(1,559)
(2,1821)
(3,2704)
(4,5210)
};
\addlegendentry{Indium}
\addplot coordinates {
(1,589)
(2,1971)
(3,2878)
};
\addlegendentry{Thallium}
\end{axis}
\end{tikzpicture}
\end{document}
Converted to PNG with the following command:
magick convert -density 120 G13_ionization_energies.pdf -quality 100 G13_ionization_energies.png