Answer
Explanation
Given:
[tex]\begin{gathered} a_1=5 \\ a_{n+1}=-2a_n+8 \end{gathered}[/tex]
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:
[tex]a_1=5[/tex]
Next, we plug in n=1 and a1=5 into the given formula to get the second term:
[tex]\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}[/tex]
Hence,
[tex]a_{2}=-2[/tex]
We plug in n=2 and a2=-2:
[tex]\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}[/tex]
We plug in n=3 and a3=12 into the given formula:
[tex]\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}[/tex]
Then, we plug in n=4 and a4=-16:
[tex]\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}[/tex]
Therefore, the answers are:
[tex]\begin{gathered} a_{1}=5 \\ a_{2}=-2 \\ a_{3}=12 \\ a_{4}=-16 \\ a_{5}=40 \end{gathered}[/tex]
