Let's write down the first five terms of the sequence [tex]\( a_n = \frac{4n}{n+5} \)[/tex].
### Finding [tex]\( a_1 \)[/tex]:
For [tex]\( n = 1 \)[/tex]:
[tex]\[
a_1 = \frac{4 \cdot 1}{1 + 5} = \frac{4}{6} = \frac{2}{3} \approx 0.6666666666666666
\][/tex]
### Finding [tex]\( a_2 \)[/tex]:
For [tex]\( n = 2 \)[/tex]:
[tex]\[
a_2 = \frac{4 \cdot 2}{2 + 5} = \frac{8}{7} \approx 1.1428571428571428
\][/tex]
### Finding [tex]\( a_3 \)[/tex]:
For [tex]\( n = 3 \)[/tex]:
[tex]\[
a_3 = \frac{4 \cdot 3}{3 + 5} = \frac{12}{8} = \frac{3}{2} = 1.5
\][/tex]
### Finding [tex]\( a_4 \)[/tex]:
For [tex]\( n = 4 \)[/tex]:
[tex]\[
a_4 = \frac{4 \cdot 4}{4 + 5} = \frac{16}{9} \approx 1.7777777777777777
\][/tex]
### Finding [tex]\( a_5 \)[/tex]:
For [tex]\( n = 5 \)[/tex]:
[tex]\[
a_5 = \frac{4 \cdot 5}{5 + 5} = \frac{20}{10} = 2
\][/tex]
Therefore, the first five terms of the sequence are:
[tex]\[
\begin{array}{l}
a_1 = 0.6666666666666666 \\
a_2 = 1.1428571428571428 \\
a_3 = 1.5 \\
a_4 = 1.7777777777777777 \\
a_5 = 2.0
\end{array}
\][/tex]
