To find the values of [tex]\( u_2 \)[/tex] and [tex]\( u_3 \)[/tex] for the given sequence defined by the recurrence relation:
[tex]\[
u_{n+1} = \frac{3}{u_n + 1}, \quad u_1 = 4
\][/tex]
we proceed as follows:
1. Calculate [tex]\( u_2 \)[/tex]:
[tex]\[
u_2 = \frac{3}{u_1 + 1}
\][/tex]
Given [tex]\( u_1 = 4 \)[/tex], substitute this value into the equation:
[tex]\[
u_2 = \frac{3}{4 + 1} = \frac{3}{5} = 0.6
\][/tex]
2. Calculate [tex]\( u_3 \)[/tex]:
[tex]\[
u_3 = \frac{3}{u_2 + 1}
\][/tex]
Substitute the value of [tex]\( u_2 \)[/tex]:
[tex]\[
u_2 = 0.6
\][/tex]
Thus,
[tex]\[
u_3 = \frac{3}{0.6 + 1} = \frac{3}{1.6} = 1.875
\][/tex]
Therefore, the values are:
[tex]\[
u_2 = 0.6
\][/tex]
[tex]\[
u_3 = 1.875
\][/tex]
