Answer
The common difference of the sum of two arithmetic sequences is the sum of the common differences of the two given arithmetic sequences.
Work Step by Step
We are given the arithmetic sequences:
$$\begin{align*}
\{a_n\}&\text{ with common difference }d\\
\{b_n\}&\text{ with common difference }f.\end{align*}$$
We have to study the sequence:
$$c_n=a_n+b_n.$$
We calculate the difference between two consecutive terms:
$$\begin{align*}
c_{k+1}-c_k&=(a_{k+1}+b_{k+1})-(a_k+b_k)\\
&=a_{k+1}+b_{k+1}-a_k-b_k\\
&=(a_{k+1}-a_k)+(b_{k+1}-b_k)\\
&=d+f.
\end{align*}$$
We got the $c_{k+1}-c_k=d+f$, therefore constant. It follows that the sequence $\{c_n\}$ is also an arithmetic sequence with common difference equal to the sum of the common differences of the two initial arithmetic sequences.
