Answer
The given sequence is an arithmetic sequence and the next two terms of the given sequence are\[-28\ \text{and}\ -35\].
Work Step by Step
To check is a sequence is an arithmetic sequence, see if the differences between two consecutive terms are equal i.e. check if\[{{a}_{n+1}}-{{a}_{n}}=d\ \ \] for all n element of 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 i.e. check if \[\frac{{{a}_{n+1}}}{{{a}_{n}}}=r\ \]for all n element of N here \[r\] is the common ratio.
When\[n=1\], for the given sequence,
\[\begin{align}
& d={{a}_{2}}-{{a}_{1}} \\
& =-7-0 \\
& =-7
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{2}}}{{{a}_{1}}} \\
& =\frac{-7}{0} \\
& =\text{Not defined}
\end{align}\]
When\[n=2\], for the given sequence,
\[\begin{align}
& d={{a}_{3}}-{{a}_{2}} \\
& =-14-\left( -7 \right) \\
& =-7
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{3}}}{{{a}_{2}}} \\
& =\frac{-14}{-7} \\
& =2
\end{align}\]
When\[n=3\], for the given sequence,
\[\begin{align}
& d={{a}_{4}}-{{a}_{3}} \\
& =-21-\left( -14 \right) \\
& =-7
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{4}}}{{{a}_{3}}} \\
& =\frac{-21}{-14} \\
& =1.5
\end{align}\]
Since, \[d=-7\ \ \forall n=1,2,3\]and \[r\]is not equal for all n element of N it implies the given sequence is an arithmetic sequence.
Use the formula \[{{a}_{n}}={{a}_{n-1}}+d\]to find the next term of the arithmetic sequence.
Put \[n=5\]in\[{{a}_{n}}={{a}_{n-1}}+d\]to find the fifth term of the given arithmetic sequence as follows:
\[\begin{align}
& {{a}_{5}}={{a}_{5-1}}+d \\
& ={{a}_{4}}+d \\
& =-21-7 \\
& =-28
\end{align}\]
Put \[n=6\]in\[{{a}_{n}}={{a}_{n-1}}+d\]to find the sixth term of the given arithmetic sequence as follows:
\[\begin{align}
& {{a}_{6}}={{a}_{6-1}}+d \\
& ={{a}_{5}}+d \\
& =-28-7 \\
& =-35
\end{align}\]
The given sequence is an arithmetic sequence and the next two terms of the given sequence are\[-28\ \text{and}\ -35\].
