Chapter 9.3, Problem 48E

Solution

To identify: Which of the following rules for $n\ge 1$ will define a geometric sequence if $a_1\ne 0$

Options (a) $a_{n+1}=a_n+3$

Options (b) $a_{n+1}=a_n-3$

Options (c) $a_{n+1}=a_n÷3$

Options (d) $a_{n+1}=a^3{}_n$

Options (e) $a_{n+1}=a_n\cdot 3^{n-1}$

Option (c) is correct option the value of is $a_{n+1}=a_n÷3$

Given information:

Rules for $n\ge 1$ will define a geometric sequence

Explanation:

Consider the given information

The recursive rule for a geometric sequence involves multiplying the previous term by a nonzero number. This matches choice C only since it can be written as:

  $a_{n+1}=a_n÷3=\frac{1}{3}{a}^n$

where $r=\frac{1}{3}$

Choices A and B are both arithmetic sequences where the common differences are $3\text{and}-3$ Choices D and E are neither. So, the correct answer is choice (C).

So, the is correct option (c).