Question

For the sequence defined by:a1=1an+1=4an+1Find:a2 = a3 = a4 =

Answer 1

Given in the question are

`\begin{gathered} a_1=1 \\ a_(n+1)=(4)/(a_n)+1 \end{gathered}`

Step 1:

To figure out the value of a2, put n=1

`\begin{gathered} a_(n+1)=(4)/(a_(n))+1 \\ a_(1+1)=(4)/(a_1)+1 \\ a_2=(4)/(a_1)+1,a_1=1 \\ a_2=(4)/(1)+1 \\ a_2=4+1 \\ a_2=5 \end{gathered}`

Hence,

`\Rightarrow a_2=5`

Step 2:

To figure out a3, put n=2

`\begin{gathered} a_(n+1)=(4)/(a_(n))+1 \\ a_(2+1)=(4)/(a_2)+1,a_2=5 \\ a_3=(4)/(5)+1 \\ a_3=(9)/(5) \end{gathered}`

Hence,

`\Rightarrow a_3=(9)/(5)`

Step 3:

To figure out the value of a4, put n=3

`\begin{gathered} a_(n+1)=(4)/(a_(n))+1 \\ a_(3+1)=(4)/(a_3)+1 \\ a_4=(4)/((9)/(5))+1 \\ a_4=4*(5)/(9)+1 \\ a_4=(20)/(9)+1 \\ a_4=(29)/(9) \end{gathered}`

Hence,

The final answer is

`\Rightarrow a_4=(29)/(9)`