Chapter 9.1, Problem 3LC

Solution

To write a recursive definition for the given sequence.

  $a_n=2a_{n-1}$

Given:

The given sequence is:

  $3,6,12,24,\dots$

Calculation:

The recursive definition describes the $n^{th}$ terms of a sequence using the previous terms of the sequence. To find the recursive definition, the first step is to identify the pattern in the consecutive terms and then write the recursive definition.

Now, for the given sequence:

  $\begin{array}{l}{a}_2=2a_1\\ \Rightarrow a_2=2\left(3\right)\\ \Rightarrow a_2=6\end{array}$

Next, third term is related as:

  $\begin{array}{l}{a}_3=2a_2\\ \Rightarrow a_3=2\left(6\right)\\ \Rightarrow a_3=12\end{array}$

And, fourth term is related as:

  $\begin{array}{l}{a}_4=2a_3\\ \Rightarrow a_4=2\left(12\right)\\ \Rightarrow a_4=24\end{array}$

Hence, from the above, the sequence follows the pattern:

  $a_n=2a_{n-1}$

Hence, the recursive definition for the given sequence is:

  $a_n=2a_{n-1}$

Here, each term is $2$ times the preceding term with an initial term $a_1=3.$