Question

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

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

Short answer

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

Step-by-step solution

Step 1. Write the given information.

The given recursively defined sequence is:

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

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

$$\begin{gathered} a_2=a_{2-1}+d \\ =a_1+d \\ =A+d \end{gathered}$$

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

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

$$\begin{gathered} a_3=a_{3-1}+d \\ =a_2+d \\ =A+d+d \\ =A+2d \\ {a}_4=a_{4-1}+d \\ =a_3+d \\ =A+2d+d \\ =A+3d \end{gathered}$$

Step 4. Find the 5th term.

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

$$\begin{gathered} a_5=a_{5-1}+d \\ =a_4+d \\ =A+3d+d \\ =A+4d \end{gathered}$$

Therefore, the first five terms are$A,A+d,A+2d,A+3d,A+4d$.