One way to analyze data is by creating a model. Models are simplified descriptions that capture some qualities of the original phenomenon. These can be used to study underlying patterns and properties of the real data. For this, consider the sequence $b_n$ given by the recurrence relation,

$b_n=\sqrt[\phantom{2}]{3}{b}_{n-1}-b_{n-2}+1875,b_2=6980,b_1=7040$

Create a graph of $b_n,$ and the graphs of $a_n$ and $b_n.$ Do you think $b_n$ is a good approximation of $a_n?$ Explain your reasoning.

What patterns do you see in $b_n?$ Do these patterns also appear in $a_n?$ Explain what it might mean for patterns to be shared or not shared between $a_n$ and $b_n$