Consider the explicit formulas for two sequences.
[tex]
\begin{array}{l}
f(n)=2^{(n-1)}-1 \\
g(n)=3 n+6
\end{array}
[/tex]
Which mathematical statement is correct?
A. [tex]$g(6)\ \textgreater \ f(6)$[/tex]
B. [tex]$f(5)\ \textless \ g(3)$[/tex]
C. [tex]$f(7)\ \textgreater \ g(10)$[/tex]
D. [tex]$g(8)=f(5)$[/tex]
Answer (1)
Calculate f ( 6 ) and g ( 6 ) and determine that f(6)"> g ( 6 ) > f ( 6 ) is false.
Calculate f ( 5 ) and g ( 3 ) and determine that f ( 5 ) < g ( 3 ) is false.
Calculate f ( 7 ) and g ( 10 ) and determine that g(10)"> f ( 7 ) > g ( 10 ) is true.
Calculate g ( 8 ) and f ( 5 ) and determine that g ( 8 ) = f ( 5 ) is false.
Conclude that the correct answer is g(10)}"> f ( 7 ) > g ( 10 ) .
Explanation
Understanding the Problem We are given two sequences defined by explicit formulas:
f ( n ) = 2 ( n − 1 ) − 1
g ( n ) = 3 n + 6
We need to determine which of the given mathematical statements is correct.
Evaluating Option A Let's evaluate each option:
Option A: f(6)"> g ( 6 ) > f ( 6 )
f ( 6 ) = 2 ( 6 − 1 ) − 1 = 2 5 − 1 = 32 − 1 = 31
g ( 6 ) = 3 ( 6 ) + 6 = 18 + 6 = 24
Is 31"> 24 > 31 ? No, so option A is false.
Evaluating Option B Option B: f ( 5 ) < g ( 3 )
f ( 5 ) = 2 ( 5 − 1 ) − 1 = 2 4 − 1 = 16 − 1 = 15
g ( 3 ) = 3 ( 3 ) + 6 = 9 + 6 = 15
Is 15 < 15 ? No, so option B is false.
Evaluating Option C Option C: g(10)"> f ( 7 ) > g ( 10 )
f ( 7 ) = 2 ( 7 − 1 ) − 1 = 2 6 − 1 = 64 − 1 = 63
g ( 10 ) = 3 ( 10 ) + 6 = 30 + 6 = 36
Is 36"> 63 > 36 ? Yes, so option C is true.
Evaluating Option D Option D: g ( 8 ) = f ( 5 )
g ( 8 ) = 3 ( 8 ) + 6 = 24 + 6 = 30
f ( 5 ) = 2 ( 5 − 1 ) − 1 = 2 4 − 1 = 16 − 1 = 15
Is 30 = 15 ? No, so option D is false.
Conclusion Therefore, the correct statement is option C: g(10)"> f ( 7 ) > g ( 10 ) .
Examples
Understanding sequences and their explicit formulas is crucial in many areas, such as predicting population growth or analyzing financial investments. For instance, if you invest money in a savings account with a fixed interest rate, the amount of money you have each year can be modeled by a sequence. By comparing different investment options, you can determine which one will yield the highest return over time, similar to how we compared the values of the sequences f(n) and g(n) for different values of n.
