Question

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

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

Short answer

The first five terms of the recursively defined sequence are$-2,0,3,7,\mathrm{and}12$.

Step-by-step solution

Step 1. Write the given information.

The given recursively defined sequence is:

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

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

The first term is$a_1=-2$.

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

$$\begin{gathered} a_2=2+a_{2-1} \\ =2+a_1 \\ =2+(-2) \\ =0 \end{gathered}$$

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

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

$$\begin{gathered} a_3=3+a_{3-1} \\ =3+a_2 \\ =3+0 \\ =3 \\ {a}_4=4+a_{4-1} \\ =4+a_3 \\ =4+3 \\ =7 \end{gathered}$$

Step 4. Find the 5th term.

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

$$\begin{gathered} a_5=5+a_{5-1} \\ =5+a_4 \\ =5+7 \\ =12 \end{gathered}$$

Therefore, the first five terms are$-2,0,3,7,12$.