Question

write a recursive formula and an explicit formula for the arithmetic sequence shown in the graph. a recursive formula is a₁ = \square, aₙ = \square. (simplify your answers.)

Explanation:

Step1: Identify \(a_1\) from the graph

The first point (when \(n = 1\)) has a \(y\)-value of \(12\), so \(a_1=12\).

Step2: Find the common difference \(d\)

Look at the next points. From \(12\) to \(8\) (when \(n = 2\)), the difference is \(8 - 12=- 4\). Check another: \(4-8 = - 4\), so \(d=-4\).

Step3: Write the recursive formula

For an arithmetic sequence, the recursive formula is \(a_1=\text{first term}\), \(a_n=a_{n - 1}+d\). Substituting \(a_1 = 12\) and \(d=-4\), we get \(a_1 = 12\), \(a_n=a_{n - 1}-4\) (since \(d=-4\)).

Answer:

\(a_1=\boldsymbol{12}\), \(a_n=\boldsymbol{a_{n - 1}-4}\)