Question

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

$a_1=-2;a_n=n+3a_{n-1}$

Short answer

The first five terms of the recursively defined sequence are$-2,-4,-9,-23,\mathrm{and}-64$.

Step-by-step solution

Step 1. Write the given information.

The given recursively defined sequence is:

$$\begin{gathered} a_1=-2 \\ {a}_n=n+3a_{n-1} \end{gathered}$$

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 $a_n=n+3a_{n-1}$to get the second term,

role="math" localid="1646730495162" $$\begin{gathered} a_2=2+3a_{2-1} \\ =2+3a_1 \\ =2+3\left(-2\right) \\ =2-6 \\ =-4 \end{gathered}$$

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

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

$$\begin{gathered} a_3=3+3a_{3-1} \\ =3+3a_2 \\ =3+3\left(-4\right) \\ =-9 \\ {a}_4=4+3a_{4-1} \\ =4+3a_3 \\ =4+3\left(-9\right) \\ =4-27 \\ =-23 \end{gathered}$$

Step 4. Find the 5th term.

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

$$\begin{gathered} a_5=5+3a_{5-1} \\ =5+3a_4 \\ =5+3\left(-23\right) \\ =5-69 \\ =-64 \end{gathered}$$

Therefore, the first five terms are$-2,-4,-9,-23,-64$.