Answer
The given sequence is an arithmetic sequence and the next two terms are\[24\ \text{and }29\].
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 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 N, here \[r\] is the ratio.
When\[n=1\], for the given sequence,
\[\begin{align}
& d={{a}_{2}}-{{a}_{1}} \\
& =9-4 \\
& =5
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{2}}}{{{a}_{1}}} \\
& =\frac{9}{4} \\
& =2.25
\end{align}\]
When\[n=2\], for the given sequence,
\[\begin{align}
& d={{a}_{3}}-{{a}_{2}} \\
& =14-9 \\
& =5
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{3}}}{{{a}_{2}}} \\
& =\frac{14}{9} \\
& =1.56
\end{align}\]
When\[n=3\], for the given sequence,
\[\begin{align}
& d={{a}_{4}}-{{a}_{3}} \\
& =19-14 \\
& =5
\end{align}\]
And,
\[\begin{align}
& r=\frac{{{a}_{4}}}{{{a}_{3}}} \\
& =\frac{19}{14} \\
& =1.35
\end{align}\]
Since, d=5, for all n=1,2,3 and r is not equal for all n in 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:
\[\begin{align}
& {{a}_{5}}={{a}_{5-1}}+d \\
& ={{a}_{4}}+d \\
& =19+5 \\
& =24
\end{align}\]
Put \[n=6\]in\[{{a}_{n}}={{a}_{n-1}}+d\]to find the sixth term of the given arithmetic sequence as follows:
\[\begin{align}
& {{a}_{6}}={{a}_{6-1}}+d \\
& ={{a}_{5}}+d \\
& =24+5 \\
& =29
\end{align}\]
The given sequence is an arithmetic sequence and the next two terms are\[24\ \text{and }29\].
