Question

find the $12^{\text{th}}$ term of the arithmetic sequence whose common difference is $d = -6$ and whose first term is $a_1 = 30$.

Explanation:

Step1: Recall the formula for the nth term of an arithmetic sequence

The formula for the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

Step2: Substitute the given values into the formula

We are given that \( a_1 = 30 \), \( d = -6 \), and \( n = 12 \). Substituting these values into the formula:
\( a_{12} = 30 + (12 - 1)(-6) \)

Step3: Simplify the expression

First, calculate \( 12 - 1 = 11 \). Then, multiply \( 11 \) by \( -6 \): \( 11 \times (-6) = -66 \). Now, add this result to \( 30 \): \( 30 + (-66) = 30 - 66 = -36 \).

Answer:

\( -36 \)