Question

How do you write the recursive formula for the following sequence $4$, $11$, $18$, $25$, $32$?

Answer 1

Step 1: Find the common difference.

Given sequence is $4$, $11$, $18$, $25$, $32$.

First term of the sequence is $a_0=4$.

The common difference between the second and the first term is $d=11-4=7$.

The common difference between the third and the second term is $d=18-11=7$.

The common difference between the fourth and the third term is $d=25-18=7$.

The common difference between the fifth and the fourth term is $d=32-25=7$.

Therefore, we can conclude that the common difference of the sequence is $7$.

Step 2: Find recursive formula.

First term of the sequence is $a_0=4$. and the common difference of the sequence is $7$.

Thus,

$$\begin{gathered} a_1=4+7 \\ \Rightarrow a_1=a_0+7 \end{gathered}$$

$$\begin{gathered} a_2=11+7 \\ \Rightarrow a_2=a_1+7 \end{gathered}$$

$$\begin{gathered} a_3=18+7 \\ \Rightarrow a_3=a_2+7 \end{gathered}$$

$$\begin{gathered} a_4=25+7 \\ \Rightarrow a_4=a_3+7 \end{gathered}$$

Therefore, the recursive formula can be written as $a_n=a_{n-1}+7$.

Hence, the recursive formula is $a_n=a_{n-1}+7$.