Write the first five terms of the sequences whose nth term is

Substituting n = 1, 2, 3, 4, 5, we obtain

$a_{1}=1 \cdot \frac{1^{2}+5}{4}=\frac{6}{4}=\frac{3}{2}$

$a_{2}=2 \cdot \frac{2^{2}+5}{4}=2 \cdot \frac{9}{4}=\frac{9}{2}$

$a_{3}=3 \cdot \frac{3^{2}+5}{4}=3 \cdot \frac{14}{4}=\frac{21}{2}$

$a_{4}=4 \cdot \frac{4^{2}+5}{4}=21$

$a_{5}=5 \cdot \frac{5^{2}+5}{4}=5 \cdot \frac{30}{4}=\frac{75}{2}$

Therefore, the required terms are $\frac{3}{2}, \frac{9}{2}, \frac{21}{2}, 21$, and $\frac{75}{2}$.