Answer
The provided statement is false. If the arithmetic sequence is\[1,\ 4,\ 7,\ 10,\ 13,\ \ldots \], common difference is \[3\].
Work Step by Step
The sequence \[1,\ 4,\ 8,\ 13,\ 19,\ 26,\ \ldots \] is an arithmetic sequence.
In the provided sequence,\[{{a}_{1}}=1,\ {{a}_{2}}=4,\ {{a}_{3}}=8,\ \ldots \]
The common difference is
\[\begin{align}
& d={{a}_{2}}-{{a}_{1}} \\
& ={{a}_{3}}-{{a}_{2}} \\
& ={{a}_{4}}-{{a}_{3}}
\end{align}\]
and
\[\begin{align}
& {{d}_{1}}={{a}_{2}}-{{a}_{1}} \\
& {{d}_{1}}=4-1 \\
& =3
\end{align}\]
and
\[\begin{align}
& {{d}_{2}}={{a}_{3}}-{{a}_{2}} \\
& =8-4 \\
& =4
\end{align}\]
Hence, \[{{d}_{1}}\ne {{d}_{2}}\](common difference is not same).
Hence, the provided sequence is not A.P.
If the arithmetic sequence is\[1,4,7,10,13,\ldots \], common difference is \[3\].
Hence, the provided statement is false.
