determine whether the sequence is arithmetic and find the common difference if it is arithmetic and if it is find the 32nd term 3,18,33,48,.

determine whether the sequence is arithmetic and find the common difference if it is arithmetic and if it is find the 32nd term 3,18,33,48,...

$$\small{\text{
we have $a_1=3, ~a_2=18,~a_3=33,~a_4=48$
}}\\\\
\small{\text{
the differences are $a_2-a_1=18-3=\mathbf{15}, ~a_3-a_2=33-18=\mathbf{15},~a_4-a_3=48-33=\mathbf{15}$
}}\\
\small{\text{
the common difference $ \mathbf{d} $ is $\mathbf{15} $
the sequence is arithmetic
}}\\\\
\small{\text{
the terms are $ a_n=a_1+(n-1)\cdot d $
}}\\\\
\small{\text{
the 32nd term is:
}}
\small{\text{$
\begin{array}{rcl}
a_{32} &=& 3+(32-1)\cdot 15\\
a_{32} &=& 3+31\cdot 15\\
\mathbf{a_{32}} &\mathbf{=}& \mathbf{468}\\
\end{array}
$}}\\\\$$