Question

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

$a_1=3;a_n=\frac{a_{n-1}}{n}$

Short answer

The first five terms of the recursively defined sequence are $3,\frac{3}{2},\frac{1}{2},\frac{1}{8},\mathrm{and}\frac{1}{40}$.

Step-by-step solution

Step 1. Write the given information.

The given recursively defined sequence is:

$$\begin{gathered} a_1=3 \\ {a}_n=\frac{a_{n-1}}{n} \end{gathered}$$

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

The first term is $\left(a_1\right)=3$.

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

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

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

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

$$\begin{gathered} a_3=\frac{a_{3-1}}{3} \\ =\frac{a_2}{3} \\ =\frac{\frac{3}{2}}{3} \\ =\frac{1}{2} \\ {a}_4=\frac{a_{4-1}}{4} \\ =\frac{a_3}{4} \\ =\frac{{\displaystyle \frac{1}{2}}}{4} \\ =\frac{1}{8} \end{gathered}$$

Step 4. Find the 5th term.

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

$$\begin{gathered} a_5=\frac{a_{5-1}}{5} \\ =\frac{a_4}{5} \\ =\frac{{\displaystyle \frac{1}{8}}}{5} \\ =\frac{1}{40} \end{gathered}$$

Therefore, the first five terms are$3,\frac{3}{2},\frac{1}{2},\frac{1}{8},\frac{1}{40}$.