The sequence

\[ 2, 2^2, 2^{2^2}, 2^{2^{2^2}}, \ldots \]

is defined by \(a_1=2 \) and \(a_{n+1}=2^{a_n} \) for all \(n\geq 1 %speech% ,n greater than or equal to 1,\).

What is the first term in the sequence greater than \(1000^{1000} %speech% ,1000 to the power of 1000,\)?

\(\Large a_4 = 2^{2^{2^2}} \vphantom{\Large a_8 = 2^{2^{2^{2^{2^{2^{2^2}}}}}}}\)\(\Large a_5 = 2^{2^{2^{2^2}}} \vphantom{\Large a_8 = 2^{2^{2^{2^{2^{2^{2^2}}}}}}}\)\(\Large a_6 = 2^{2^{2^{2^{2^2}}}} \vphantom{\Large a_8 = 2^{2^{2^{2^{2^{2^{2^2}}}}}}}\)\(\Large a_7 = 2^{2^{2^{2^{2^{2^2}}}}} \vphantom{\Large a_8 = 2^{2^{2^{2^{2^{2^{2^2}}}}}}}\)\(\Large a_8 = 2^{2^{2^{2^{2^{2^{2^2}}}}}}\)