Chapter 7.8, Problem 61MR

To determine

To write explicit and a recursive formula for a geometric sequence $-7,-3,1,\mathrm{5......}$ .

Answer to Problem 61MR

The explicit rule is $a_n=4n-11$ and recursive rule is $a_1=-7,a_n=a_{n-1}+4.$

Explanation of Solution

Given information :

Given arithmetic sequence $-7,-3,1,\mathrm{5......}$

The given arithmetic sequence $-7,-3,1,\mathrm{5......}$

Where first term $a_1=-7$ and common difference $d=-3-(-7)=-3+7=4$

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

  $\begin{array}{l}\Rightarrow a_n=\left(-7\right)+\left(n-1\right)4\\ =-7+4n-4\\ \therefore a_n=4n-11\end{array}$

Thus, recursive rule for an arithmetic sequence is $a_n=a_{n-1}+d$

  $\Rightarrow a_n=a_{n-1}+4$

Hence, the explicit rule is $a_n=4n-11$ and recursive rule is $a_1=-7,a_n=a_{n-1}+4.$