So the points given by the table define an arithmetic sequence i.e. a sequence where the difference between consecutive terms remains constant and is known as the common difference. The general formula for this type of sequences is:
[tex]a_n=a_1+(n-1)\cdot d[/tex]Where n is a positive integer and d is the common difference.
The points given by the question have the following form:
[tex](n,a_n)[/tex]Which means that the first four terms of the sequence are:
[tex]\begin{gathered} a_1=-6 \\ a_2=-9 \\ a_3=-12 \\ a_4=-15 \end{gathered}[/tex]In order to find d we just need to take a term and substract the previous one:
[tex]a_2-a_1=a_3-a_2=a_4-a_3=-3[/tex]So d=-3 and the answer to the first empty box is -3.
Now we have to find the explicit formula. We already found d so for now we have:
[tex]\begin{gathered} a_n=a_1+(n-1)\cdot(-3) \\ a_n=a_1-3(n-1) \end{gathered}[/tex]And we also know the value of a₁, it's -6 so we get:
[tex]a_n=-6-3(n-1)[/tex]And that's the answer to the second box.
