Questions: Write a recursive formula and an explicit formula for the arithmetic sequence shown in the graph. A recursive formula is a1= , an= . (Simplify your answers.)

Solution Steps

The first term, $a_1$, corresponds to the y-value when $x=1$. From the graph, we can see that $a_1 = -12$.

To find the common difference, _d_, we can subtract the y-value at $x=1$ from the y-value at $x=2$. $d = -8 - (-12) = -8 + 12 = 4$

A recursive formula defines each term of a sequence based on the preceding term. The first term is given as $a_1$. The subsequent terms, $a_n$, are defined by adding the common difference, _d_, to the previous term, $a_{n-1}$.

In this case, $a_1 = -12$ and $d=4$, so the recursive formula is: $a_1 = -12$, $a_n = a_{n-1} + 4$

An explicit formula defines each term of a sequence directly in terms of _n_. The general form of an explicit formula for an arithmetic sequence is: $a_n = a_1 + (n-1)d$

In this case, $a_1 = -12$ and $d=4$, so the explicit formula is: $a_n = -12 + (n-1)4$ $a_n = -12 + 4n - 4$ $a_n = 4n - 16$

Final Answer