Question

In Problems 37–50, a sequence is defined recursively. Write down the first five terms.

$a_1=1;a_2=2;a_n=a_{n-1}·a_{n-2}$

Short answer

The first five terms of the recursively defined sequence are$1,2,2,4,\mathrm{and}8$.

Step-by-step solution

Step 1. Write the given information.

The given recursively defined sequence is:

$$\begin{gathered} a_1=1 \\ {a}_2=2 \\ {a}_n=a_{n-1}·a_{n-2} \end{gathered}$$

Step 2. Determine the first and second terms and find out the third term.

The first term is $\left(a_1\right)=1$.

The second term is role="math" localid="1646733478804" $\left(a_2\right)=2$

Now substitute 3 for nin the given formula $a_n=a_{n-1}·a_{n-2}$to get the third term,

$$\begin{gathered} a_3=a_{3-1}·a_{3-2} \\ =a_2·a_1 \\ =1·2 \\ =2 \end{gathered}$$

Step 3. Find the 4th and 5th terms.

Similarly, substitute 4 and 5 for nin the given formula to get the fourth and fifth term,

$$\begin{gathered} a_4=a_{4-1}·a_{4-2} \\ =a_3·a_2 \\ =2·2 \\ =4 \\ {a}_5=a_{5-1}·a_{5-2} \\ =a_4·a_3 \\ =4·2 \\ =8 \end{gathered}$$

Therefore, the first five terms are$1,2,2,4,8$.