Question

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

$a_1=A;a_n=r{a}_{n-1},r\ne 0$

Short answer

The first five terms of the recursively defined sequence are$A,rA,r^{2}A,r^{3}A,\mathrm{and}{r}^{4}A$.

Step-by-step solution

Step 1. Write the given information.

The given recursively defined sequence is:

$$\begin{gathered} a_1=A \\ {a}_n=r{a}_{n-1},r\ne 0 \end{gathered}$$

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

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

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

$$\begin{gathered} a_2=r{a}_{2-1} \\ =r{a}_1 \\ =rA \end{gathered}$$

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

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

$$\begin{gathered} a_3=r{a}_{3-1} \\ =r{a}_2 \\ =r·rA \\ =r^{2}A \\ {a}_4=r{a}_{4-1} \\ =r{a}_3 \\ =r·r^{2}A \\ =r^{3}A \end{gathered}$$

Step 4. Find the 5th term.

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

$$\begin{gathered} a_5=r{a}_{5-1} \\ =r{a}_4 \\ =r·r^{3}A \\ =r^{4}A \end{gathered}$$

Therefore, the first five terms are$A,rA,r^{2}A,r^{3}A,r^{4}A$.