Question

I have a precalculus question about sequences with the picture included

Answer 1

Step-by-step explanation

Given:

`\begin{gathered} a_1=5 \\ a_(n+1)=-2a_n+8 \end{gathered}`

To determine the first five terms of the given recursive defined sequence, we follow the process as shown below:

The first term is given, so:

`a_1=5`

Next, we plug in n=1 and a1=5 into the given formula to get the second term:

`\begin{gathered} a_(n+1)=-2a_(n)+8 \\ a_(1+1)=-2a_1+8 \\ Simplify \\ a_2=-2a_1+8 \\ a_2=-2(5)+8 \\ a_2=-2 \end{gathered}`

Hence,

`a_(2)=-2`

We plug in n=2 and a2=-2:

`\begin{gathered} a_(n+1)=-2a_(n)+8 \\ a_(2+1)=-2a_2+8 \\ Simplify \\ a_3=-2(-2)+8 \\ a_3=12 \end{gathered}`

We plug in n=3 and a3=12 into the given formula:

`\begin{gathered} a_(n+1)=-2a_(n)+8 \\ a_(3+1)=-2a_3+8 \\ Simplify \\ a_4=-2(12)+8 \\ a_4=-16 \end{gathered}`

Then, we plug in n=4 and a4=-16:

`\begin{gathered} a_(n+1)=-2a_(n)+8 \\ a_(4+1)=-2a_4+8 \\ Simplify \\ a_5=-2(-16)+8 \\ a_5=40 \end{gathered}`

Therefore, the answers are:

`\begin{gathered} a_(1)=5 \\ a_(2)=-2 \\ a_(3)=12 \\ a_(4)=-16 \\ a_(5)=40 \end{gathered}`