Answer
\begin{array}{lclcl}
a_1=2&\\
a_2=5&\\
a_3=47&\\
a_4=4415\\
\end{array}
Work Step by Step
Given $$ a_1=2, \ \ \ a_{n+1}=2a_n^2-3, \ \ \ \ \ \ n= 1,2,3, \dots
$$
So, we get
\begin{array}{|l|l|}\hline
& n& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a_n \\ \\ \hline
\text{First Term} & 1 & a_1=2 \\ \\ \hline
\text{Second Term} & 2 & a_2=2a_1^2-3 =2 \times 2^2 -3=8-3=5\\ \\ \hline
\text{Third Term} & 3 & a_3=2a_2^2-3 =2 \times 5^2 -3=50-3=47 \\ \\ \hline
\text{Fourth Term} & 4 & a_4=2a_3^2-3 =2 \times 47^2 -3=4418-3=4415\\ \\ \hline
\end{array}
