Consider the following empty table for an arithmetic sequence defined as $f(n)=12+9(n-1)$ and a geometric sequence defined as $g(n)=$ $64 \cdot\left(\frac{1}{2}\right)^{n}$. a) Arithmetic b) Geometric \begin{tabular}{|c|c|c|c|c|c|} \hline$n$ & 1 & 2 & 3 & 4 & 5 \\ \hline$f(n)$ & & & & & \\ \hline$g(n)$ & & & & & \\ \hline \end{tabular} Enter the missing values for row a), in order, separated by a comma. $12,21,30,39,48$ $12,21,30,39,48$ Enter the missing values for row b), in order, separated by a comma. Which is the greater value $f(100)$ or $g(100)$ and why? $f(100)$; because the arithmetic sequence is increasing while the geometric sequence is decreasing. $g(100)$; because a geometric sequence is multiplying each value which will eventually exceed the addition of an arithmetic sequence.
Solution
Step 1 ：Define the arithmetic sequence as \(f(n)=12+9(n-1)\) and the geometric sequence as \(g(n)=64 \cdot\left(\frac{1}{2}\right)^{n}\).
Step 2 ：Calculate the first 5 terms of the arithmetic sequence by substituting the values of n from 1 to 5 into the formula. The results are \(12, 21, 30, 39, 48\).
Step 3 ：Calculate the first 5 terms of the geometric sequence by substituting the values of n from 1 to 5 into the formula. The results are \(32.0, 16.0, 8.0, 4.0, 2.0\).
Step 4 ：Calculate the 100th term of both sequences using their respective formulas. The 100th term of the arithmetic sequence is \(f(100) = 903\) and the 100th term of the geometric sequence is \(g(100) = 5.048709793414476e-29\).
Step 5 ：Compare the 100th term of both sequences. The greater value is \(f(100)\) because \(903 > 5.048709793414476e-29\).
Step 6 ：Final Answer: The missing values for row a) are \(\boxed{12, 21, 30, 39, 48}\). The missing values for row b) are \(\boxed{32.0, 16.0, 8.0, 4.0, 2.0}\). The greater value between \(f(100)\) and \(g(100)\) is \(\boxed{f(100)}\).
