Answer
a) Yes
b) Yes
Work Step by Step
a) We are given the arithmetic sequence with common difference $d$.
We have to study the sequence:
$$b_n=a_n+2.$$
We calculate the difference between two consecutive terms:
$$\begin{align*}
b_{k+1}-b_k&=(a_{k+1}+2)-(a_k+2)\\
&=a_{k+1}+2-a_k-2\\
&=a_{k+1}-a_k\\
&=d.
\end{align*}$$
We got the $b_{k+1}-b_k=d$, therefore constant. It follows that the sequence $\{b_n\}$ is also an arithmetic sequence with common difference equal to the common difference of the initial arithmetic sequence.
b) We are given the geometric sequence with common ratio $r$.
We have to study the sequence:
$$b_n=5a_n.$$
We calculate the ratio between two consecutive terms:
$$\begin{align*}
\dfrac{b_{k+1}}{b_k}&=\dfrac{5a_{k+1}}{5a_k}\\
&=\dfrac{a_{k+1}}{a_k}\\
&=r.
\end{align*}$$
We got that $\dfrac{b_{k+1}}{b_k}=r$, therefore constant. It follows that the sequence $\{b_n\}$ is also a geometric sequence with common ratio equal to the common ratio of the initial geometric sequence.
