Chapter 6.6, Problem 2E

To determine

To determine which rule does not belong among other three given rules

Answer to Problem 2E

Rule $a_n=6n-2$ does not belong among other three because $a_n=6n-2$ is explicit rule while other three are recursive rules

Explanation of Solution

Given information:

Given four rules:

  $\begin{array}{l}{a}_1=-1,a_n=5a_{n-1}\\ a_n=6n-2\\ a_1=-3,a_n=a_n+1\\ a_1=9,a_n=4a_{n-1}\end{array}$

Concept used:

Recursive equation for an arithmetic sequence is

  $a_n=a_{n-1}+d\text{ (where d is common difference)}$

Recursive equation for a geometric sequence is

  $a_n=r.a_{n-1}\text{ (where r is common ratio)}$

  $\begin{array}{l}\text{Explicit rule equation gives }{a}_n\text{ as a }\\ \text{function of the terms position number}\\ n\text{ in the sequence}\text{.}\end{array}$

Given four rules are

  $\begin{array}{l}{a}_1=-1,a_n=5a_{n-1}\\ a_n=6n-2\\ a_1=-3,a_n=a_n+1\\ a_1=9,a_n=4a_{n-1}\end{array}$

Compare given equation with recursive equation for an arithmetic and geometric sequence

Clearly,

Equations that represents recursive rule are:

  $\begin{array}{l}{a}_n=5a_{n-1}\text{ (with 5 as common ratio)}\\ a_n=a_n+1\text{ (with 1 as common difference)}\\ a_n=4a_{n-1}\text{ (with 4 as common ratio)}\end{array}$

Therefore,

Rule $a_n=6n-2$ does not belong among other three because $a_n=6n-2$ is representing explicit rule while other three are representing recursive rules