Question

at the local grocery store, cans of beans are arranged in rows. the number of cans in each row forms an arithmetic sequence. there are 105 cans in row 1, 96 cans in row 2, 87 cans in row 3, and so on. one row contains 42 cans. which row is this?
○ row 7
○ row 9
○ row 8
○ row 10

Explanation:

Step1: Identify arithmetic sequence parameters

The first term \(a_1 = 105\). The common difference \(d\) is \(96 - 105=-9\) (or \(87 - 96 = -9\)). The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\).

Step2: Substitute values into the formula

We know \(a_n = 42\), \(a_1=105\), \(d=-9\). Substitute into \(a_n=a_1+(n - 1)d\):
\[
42=105+(n - 1)(-9)
\]

Step3: Solve for \(n\)

First, simplify the equation:
\[
42=105-9n + 9
\]
\[
42=114-9n
\]
Subtract 114 from both sides:
\[
42-114=-9n
\]
\[
-72=-9n
\]
Divide both sides by -9:
\[
n = 8
\]

Answer:

row 8