Chapter 6.6, Problem 24E

To determine

To write an explicit rule for the recursive rule $a_1=8,a_n=a_{n-1}-12$ .

Answer to Problem 24E

The explicit rule for the given recursive rule is $20-12n$ .

Explanation of Solution

Given information:

Given recursive rule $a_1=8,a_n=a_{n-1}-12$ .

The given recursive rule $a_1=8,a_n=a_{n-1}-12$ represents an arithmetic sequence with first term $a_1=8$ and common difference $d=-12$ .

Thus, the explicit rule or $n^{th}\text{term}$ for an arithmetic sequence is $a_n=a_1+\left(n-1\right)d$

  $\begin{array}{l}\Rightarrow a_n=8+\left(n-1\right)\left(-12\right)\\ =8-12n+12\\ \therefore a_n=20-12n\end{array}$

Hence, the explicit rule for the given recursive rule is $a_n=3n-6$ .