Question

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

Answer 1

Solution:

Given in the question are

$\begin{gathered} a_1=1 \\ a_{n+1}=\frac{4}{a_n}+1 \end{gathered}$

Step 1:

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

$\begin{gathered} a_{n+1}=\frac{4}{a_{n}}+1 \\ a_{1+1}=\frac{4}{a_1}+1 \\ a_2=\frac{4}{a_1}+1,a_1=1 \\ a_2=\frac{4}{1}+1 \\ a_2=4+1 \\ a_2=5 \end{gathered}$

Hence,

$\Rightarrow a_2=5$

Step 2:

To figure out a3, put n=2

$\begin{gathered} a_{n+1}=\frac{4}{a_{n}}+1 \\ a_{2+1}=\frac{4}{a_2}+1,a_2=5 \\ a_3=\frac{4}{5}+1 \\ a_3=\frac{9}{5} \end{gathered}$

Hence,

$\Rightarrow a_3=\frac{9}{5}$

Step 3:

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

$\begin{gathered} a_{n+1}=\frac{4}{a_{n}}+1 \\ a_{3+1}=\frac{4}{a_3}+1 \\ a_4=\frac{4}{\frac{9}{5}}+1 \\ a_4=4\times\frac{5}{9}+1 \\ a_4=\frac{20}{9}+1 \\ a_4=\frac{29}{9} \end{gathered}$

Hence,

The final answer is

$\Rightarrow a_4=\frac{29}{9}$