The explicit formula of an arithmetic sequence combines the information provided by the two equations of the recursive form into a single equation.
Explicit:& a_n= a_1+(n-1) d; [0.8em]
Recursive:& a_1= a_1,
& a_n=a_(n-1)+ d
In these formulas, d is the common difference and a_1 is the first term. Looking at the given explicit formula, we can identify the common difference d as the coefficient of the n-term. Then we will calculate the value of the first term a_1.
a_n= - 4n+2
We can see that - 4 is the common difference. We can calculate the first term by substituting n=1 into the given explicit formula.
a_1=- 4(1)+2=- 4 +2= - 2
Now we have enough information to form a recursive formula for this sequence.
\begin{aligned}
a_1&={\color{#0000FF}{\text{-} 2}}, \ \ a_n=a_{n-1}+({\color{#009600}{\text{-} 4}}) \\
& \quad \quad \quad \quad \Updownarrow \\
a_1&=\text{-} 2, \ \ a_n=a_{n-1}- 4
\end{aligned}
