The sequence given can be seen as:
1, 4, 16, .......
From this three terms we notice that the terms of this sequence are obtain by multiplying the previous term by 4, this means that this a geometric sequence.
We know that a geometric series is given by:
[tex]a_n=ar^{n-1}[/tex]
where a is the first term and r is the common ratio. In this case the common ratio is 4, then the sequence is:
[tex]a_n=4^{n-1}[/tex]
This gives the total number of tiles in the nth figure, then we can rewrite as:
[tex]S(n)=4^{n-1}[/tex]
