Certainly! Let's go through the values step by step. These are the fractions of atoms that are still radioactive after [tex]\( n \)[/tex] half-life cycles:
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
A = 0.5
\][/tex]
2. For [tex]\( n = 2 \)[/tex]:
[tex]\[
B = 0.25
\][/tex]
3. For [tex]\( n = 3 \)[/tex]:
[tex]\[
C = 0.125
\][/tex]
4. For [tex]\( n = 6 \)[/tex]:
[tex]\[
D = 0.015625
\][/tex]
5. For [tex]\( n = 8 \)[/tex]:
[tex]\[
E = 0.00390625
\][/tex]
Now, let's fill in the table with these values.
[tex]\[
\begin{array}{l}
A = 0.5 \\
B = 0.25 \\
C = 0.125 \\
D = 0.015625 \\
E = 0.00390625
\end{array}
\][/tex]
[tex]\[
\begin{tabular}{|c|c|}
\hline \begin{tabular}{c}
Time-Half-Life \\
Cycles, $n$
\end{tabular} & $0.5^{ n }$ \\
\hline Initial & 1 \\
\hline 1 & 0.5 \\
\hline 2 & 0.25 \\
\hline 3 & 0.125 \\
\hline 4 & 0.0625 \\
\hline 5 & 0.03125 \\
\hline 6 & 0.015625 \\
\hline 7 & 0.0078125 \\
\hline 8 & 0.00390625 \\
\hline
\end{tabular}
\][/tex]
This completes the table and provides the fractions of radioactive atoms remaining after each specified number of half-life cycles.
