Question

Write the first five terms of the sequence, whose $n^{th}$ term is $a_n=n\frac{n^2+5}{4}$.

Answer 1

Finding sequence
Given, $a_n=n\frac{n^2+5}{4},n\in N$
Substituting $n=1,2,3,4$ and $5$ in $a_n$, we get

$a_1=1\cdot \frac{1^2+5}{4}$
$\Rightarrow a_1=1\times \frac{6}{4}$
$\Rightarrow a_1=\frac{3}{2}$

$a_2=2\cdot \frac{2^2+5}{4}$
$\Rightarrow a_2=2\times \frac{4+5}{4}$
$\Rightarrow a_2=\frac{9}{2}$

$a_3=3\cdot \frac{3^2+5}{4}$
$\Rightarrow a_3=3\times \frac{9+5}{4}$
$\Rightarrow a_3=3\times \frac{14}{4}$
$\Rightarrow a_3=\frac{21}{2}$

$a_4=4\cdot \frac{4^2+5}{4}$
$\Rightarrow a_4=4\times \frac{16+5}{4}$
$\Rightarrow a_4=21$

$a_5=5\cdot \frac{5^2+5}{4}$
$\Rightarrow a_5=5\times \frac{25+5}{4}$
$\Rightarrow a_5=\frac{75}{2}$

Hence, the first five terms of the sequence are $\frac{5}{2},\frac{9}{2},\frac{21}{2},21,\frac{75}{2}$.