Chapter 6.6, Problem 3E

To determine

To determine whether the recursive rule represents an arithmetic sequence or a geometric sequence

Answer to Problem 3E

The recursive rule representing $a_1=2,a_n=7a_{n-1}$ is a geometric sequence with common ratio $7$

Explanation of Solution

Given information:

Given $a_1=2,a_n=7a_{n-1}$

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

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

Given $a_1=2,a_n=7a_{n-1}$

Clearly,

Recursive rule, $a_n=7a_{n-1}$ is of the form $a_n=r.a_{n-1}\text{ (where r is common ratio)}$

Therefore,

The recursive rule represents $a_1=2,a_n=7a_{n-1}$ is a geometric sequence with common ratio $7$