Question

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

$a_1=5;a_n=2a_{n-1}$

Short answer

The first five terms of the recursively defined sequence are$5,10,20,40,\mathrm{and}80$.

Step-by-step solution

Step 1. Write the given information.

The given recursively defined sequence is:

$$\begin{gathered} a_1=5 \\ {a}_n=2a_{n-1} \end{gathered}$$

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

The first term is $\left(a_1\right)=5$.

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

$$\begin{gathered} a_2=2a_{2-1} \\ =2a_1 \\ =2\left(5\right) \\ =10 \end{gathered}$$

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

Similarly substitute 3, 4 for nin the given formula to get the third and fourth term,

$$\begin{gathered} a_3=2a_{3-1} \\ =2a_2 \\ =2\left(10\right) \\ =20 \\ {a}_4=2a_{4-1} \\ =2a_3 \\ =2\left(20\right) \\ =40 \end{gathered}$$

Step 4. Find the 5th term.

Now substitute 5 for nin the given formula to get the 5th term,

$$\begin{gathered} a_5=2a_{5-1} \\ =2a_4 \\ =2\left(40\right) \\ =80 \end{gathered}$$

Therefore, the first five terms are$5,10,20,40,80$.