To determine the value of [tex]\( t_3 \)[/tex] given the recurrence relation [tex]\( t_n = \frac{t_{n-1}}{n} \)[/tex] with the initial term [tex]\( t_1 = 1 \)[/tex], follow these steps:
1. Calculate [tex]\( t_2 \)[/tex]:
Using the recurrence relation for [tex]\( n = 2 \)[/tex],
[tex]\[
t_2 = \frac{t_1}{2}
\][/tex]
Substitute [tex]\( t_1 = 1 \)[/tex],
[tex]\[
t_2 = \frac{1}{2}
\][/tex]
2. Calculate [tex]\( t_3 \)[/tex]:
Using the recurrence relation for [tex]\( n = 3 \)[/tex],
[tex]\[
t_3 = \frac{t_2}{3}
\][/tex]
Substitute [tex]\( t_2 = \frac{1}{2} \)[/tex],
[tex]\[
t_3 = \frac{\frac{1}{2}}{3} = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}
\][/tex]
Therefore, the value of [tex]\( t_3 \)[/tex] is [tex]\( \frac{1}{6} \)[/tex], which corresponds to option 2.
