Answer
The given sequence is a geometric sequence and the next two terms of the given sequence are\[162\ \text{and }486\].
Work Step by Step
To check is a sequence is an arithmetic sequence, see if the differences between two consecutive terms are equal. That is, check if {a}_{n+1}-{a}_{n}=d for all n in N, here \[d\] is the common difference.
To check if a sequence is a geometric sequence, see if the ratio between two consecutive terms is equal. That is, check if {{a}_{n+1}}/{{a}_{n}}=r for all n in Nu, here \[r\] is the common ratio.
When\[n=1\], for the given sequence,
\[\begin{align}
& d={{a}_{2}}-{{a}_{1}} \\
& =6-2 \\
& =4
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{2}}}{{{a}_{1}}} \\
& =\frac{6}{2} \\
& =3
\end{align}\]
When\[n=2\], for the given sequence,
\[\begin{align}
& d={{a}_{3}}-{{a}_{2}} \\
& =18-6 \\
& =12
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{3}}}{{{a}_{2}}} \\
& =\frac{18}{6} \\
& =3
\end{align}\]
When\[n=3\], for the given sequence,
\[\begin{align}
& d={{a}_{4}}-{{a}_{3}} \\
& =54-18 \\
& =36
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{4}}}{{{a}_{3}}} \\
& =\frac{54}{18} \\
& =3
\end{align}\]
Since, \[r=3\ \ \forall n=1,2,3\]and \[d\]is not equal for all n in N, it implies the given sequence is a geometric sequence.
Use the formula \[{{a}_{n}}={{a}_{1}}{{r}^{n-1}}\]to find the next term of the arithmetic sequence.
Put \[n=5\]in\[{{a}_{n}}={{a}_{1}}{{r}^{n-1}}\]to find the fifth term of the given arithmetic sequence as follows:
\[\begin{align}
& {{a}_{5}}={{a}_{1}}{{r}^{5-1}} \\
& =2\cdot {{3}^{5-1}} \\
& =2\cdot {{3}^{4}} \\
& =162
\end{align}\]
Put \[n=6\]in\[{{a}_{n}}={{a}_{1}}{{r}^{n-1}}\]to find the sixth term of the given arithmetic sequence as follows:
\[\begin{align}
& {{a}_{6}}={{a}_{1}}{{r}^{6-1}} \\
& =2\cdot {{3}^{6-1}} \\
& =2\cdot {{3}^{5}} \\
& =482
\end{align}\]
The given sequence is a geometric sequence and the next two terms of the given sequence are\[162\ \text{and }486\].
