Question

what is the linear function equation for the arithmetic sequence $a_n = -12 + (n - 1) \cdot (-5)$? (1 point) \bigcirc $y = -5x - 7$ \bigcirc $y = -12x - 5$ \bigcirc $y = -5x - 12$ \bigcirc $y = -7x - 5$

Explanation:

Step1: Expand the arithmetic sequence formula

Given \( a_n = -12+(n - 1)\cdot(-5) \), first expand the term \( (n - 1)\cdot(-5) \). Using the distributive property, \( (n - 1)\cdot(-5)=-5n + 5 \). So the formula becomes \( a_n=-12-5n + 5 \).

Step2: Simplify the arithmetic sequence formula

Combine like terms in \( -12-5n + 5 \). The constant terms \( -12 \) and \( 5 \) can be combined: \( -12 + 5=-7 \). So \( a_n=-5n-7 \).

Step3: Relate to linear function

In a linear function \( y = mx + b \), we can think of \( a_n \) as \( y \) and \( n \) as \( x \). So the linear function corresponding to the arithmetic sequence is \( y=-5x - 7 \).

Answer:

\( y = -5x - 7 \) (which corresponds to the first option: \( y = -5x - 7 \))