Two sequences $a$ and $b$ for which $\Delta a_n = \Delta b _n$

Question

Find two different sequences $a$ and $b$ for which $\Delta a_n = \Delta b_n$ for all of $n$.

This is my first time doing recurrence relations, so if anyone could provide some thorough and clear hints/examples, it would be great.

Answer 1

Hint: $\Delta a_n = \Delta b _n$ implies $\Delta (a_n - b _n)=0$. What does this imply about $a_n - b _n$?

Answer 2

If a step from a(n) to a(n+1) size delta is always the same as a the step from b(n) to b(n+1), then starting from a(1) and going k steps further to a(k+1) should also be true for the b-sequence: from b(1) to b(k+1). So, in a plot, they rise in parallel. How does one translate that analytically?