Answer
a) yes it will be arithmetic sequence too
b) yes it will be geometric sequence too
Work Step by Step
If
$a_{1},a_{2},a_{3}\ldots $ an arithmetic sequence then $a_{3}-a_{2}=a_{2}-a_{1}$
In order $a_{1}+2,a_{2}+2,a_{3}+2\dots $ be an arithmetic sequence
$\left( a_{3}+2\right) -\left( a_{2}+2\right) =\left( a_{2}+2\right) -\left( a_{1}+2\right) $
$\left( a_{3}+2\right) -\left( a_{2}+2\right) =\left( a_{2}+2\right) -\left( a_{1}+2\right) \Rightarrow a_{3}-a_{2}=a_{2}-a_{1}$ so this is arithmetic sequence too
If $5a_{1},5a_{2},5a_{3}\ldots $ is a geometric sequence then
$\left( 5a_{3}\right) \times \left( 5a_{1}\right) =\left( 5a_{2}\right) ^{2}\Rightarrow a_{3}\times a_{1}=a^{2}_{2}$ and we already know this is true becouse
$a_{1},a_{2},a_{3}\dots $ is a geometric sequence
