Answer
$a_1=\frac{3}{2}$
$a_2=\frac{3}{2}$
$a_3=\frac{9}{8}$
$a_4=\frac{3}{4}$
$a_5=\frac{15}{32}$
$S_{5}=\frac{171}{32}=5.34375$
Work Step by Step
We have to find first five terms of the sequence where
$a_n=\frac{1}{2^n}\log1000^n=\frac{1}{2^n}\log 10^{3n}=\frac{3n}{2^n}$
and $S_{5}$.
Calculate the first $5$ terms using the formula for the general term:
$$\begin{align*}
a_1&=\frac{3\cdot 1}{2^1}=\frac{3}{2}\\
a_2&=\frac{3\cdot 2}{2^2}=\frac{3}{2}\\
a_3&=\frac{3\cdot 3}{2^3}=\frac{9}{8}\\
a_4&=\frac{3\cdot 4}{2^4}=\frac{3}{4}\\
a_5&=\frac{3\cdot 5}{2^5}=\frac{15}{32}
\end{align*}$$
Determine $S_5$:
$$\begin{align*}
S_{5}&=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}\\
&=\frac{3}{2}+\frac{3}{2}+\frac{9}{8}+\frac{3}{4}+\frac{15}{32}\\
&=3+\frac{51}{32}+\frac{3}{4}\\
&=3+\frac{75}{32}\\
&=\frac{171}{32}
\end{align*}$$
