Question

Write both the explicit and recursive equations for this sequence: 7, 10, 13, 16,…. Show yourwork/explain your thinking for how you get each component of both of the equations.

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given sequence

`7,10,13,16,...`

STEP 2: Find the common difference

It can be seen that the given sequence is an arithmetic sequence, therefore, the common difference is:

`\begin{gathered} d=T_2-T_1=T_3-T_2=T_4-T_3 \\ T_1=a=7 \\ T_2=10 \\ T_3=13 \\ T_4=16 \\ \\ d=10-7=13-10=16-13=3 \\ \end{gathered}`

STEP 3: Get the explicit equation

Using the equation for getting the nth term of an arithmetic sequence which is given below:

`T_n=a+(n-1)d`

a = 7, d =3,

hence the explicit equation becomes:

`\begin{gathered} T_n=7+(n-1)3 \\ T_n=7+3n-3 \\ T_n=4+3n \end{gathered}`

STEP 4: Get the recursive equation

`\begin{gathered} T_1=a=7 \\ T_2=T_1+d \\ T_3=T_2+d \\ T_n=T_(n-1)+d \end{gathered}`

Therefore the equations are given below:

Explicit equation

`T_n=4+3n`

Recursive equation