Writing and Using Explicit Rules for Arithmetic Sequences

Explicit equations for arithmetic sequences follow a specific format. $$\begin{array}{c}{a}_n=a_1+(n-1)d\end{array}$$ In this form, $a_1$ is the first term in a given sequence, $d$ is the common difference from one term to the next, and $a_n$ is the $n^{\text{th}}$ term in the sequence. For this exercise, the first term is $a_1=\text{-}6.$ Let's observe the other terms to determine the common difference $d.$ $$\begin{array}{c}\text{-}6\stackrel{\text{-}3}{\to }\text{-}9\stackrel{\text{-}3}{\to }\text{-}12\stackrel{\text{-}3}{\to }\text{-}15\dots \end{array}$$ As we can see $d=\text{-}3.$ By substituting these two values we found into the explicit equation and simplifying, we can find the formula for this sequence. This equation can be used to find any term in the given sequence. To find $a_{10},$ the ${10}^{\text{th}}$ term in the sequence, we substitute $10$ for $n.$ The $10^{\text{th}}$ term in the sequence is $\text{-}33.$