20. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ be terms of an arithmetic sequence with positive terms and common difference $\mathrm { d } ( d \neq 0 )$ (1) Prove that $2 ^ { a _ { 1 } } , 2 ^ { a _ { 2 } } , 2 ^ { a _ { 3 } } , 2 ^ { a _ { 4 } }$ form a geometric sequence in order (2) Do there exist $a _ { 1 } , d$ such that $a _ { 1 } , a _ { 2 } { } ^ { 2 } , a _ { 3 } { } ^ { 3 } , a _ { 4 } { } ^ { 4 }$ form a geometric sequence in order? Explain your reasoning (3) Do there exist $a _ { 1 } , d$ and positive integers $n , k$ such that $a _ { 1 } { } ^ { n } , a _ { 2 } { } ^ { n + k } , a _ { 3 } { } ^ { n + 3 k } , a _ { 4 } { } ^ { n + 5 k }$ form a geometric sequence in order? Explain your reasoning
Supplementary Problems
20. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ be terms of an arithmetic sequence with positive terms and common difference $\mathrm { d } ( d \neq 0 )$\\
(1) Prove that $2 ^ { a _ { 1 } } , 2 ^ { a _ { 2 } } , 2 ^ { a _ { 3 } } , 2 ^ { a _ { 4 } }$ form a geometric sequence in order\\
(2) Do there exist $a _ { 1 } , d$ such that $a _ { 1 } , a _ { 2 } { } ^ { 2 } , a _ { 3 } { } ^ { 3 } , a _ { 4 } { } ^ { 4 }$ form a geometric sequence in order? Explain your reasoning\\
(3) Do there exist $a _ { 1 } , d$ and positive integers $n , k$ such that $a _ { 1 } { } ^ { n } , a _ { 2 } { } ^ { n + k } , a _ { 3 } { } ^ { n + 3 k } , a _ { 4 } { } ^ { n + 5 k }$ form a geometric sequence in order? Explain your reasoning
\section*{Supplementary Problems}