Respuesta :
Given the situation to model the arithmetic and the geometric sequences.
Imagine you are four years old. A rich aunt wants to provide for your future. She has offered to do one of two things.
Option 1: She would give you $1000.50 a year until you are twenty-one.
This option represents the arithmetic sequence
The first term = a = 1000.50
The common difference = d = 1000.50
The general formula will be as follows:
[tex]\begin{gathered} a_n=a+d(n-1) \\ a_n=1000.50+1000.50(n-1) \\ \end{gathered}[/tex]Simplify the expression:
[tex]a_n=1000.50n[/tex]Option 2: She would give you $1 this year, $2 next year, and so on, doubling the amount each year until you were 21.
This option represents the geometric sequence
The first term = a = 1
The common ratio = r = 2/1 = 2
The general formula will be as follows:
[tex]\begin{gathered} a_n=a\cdot r^{n-1} \\ a_n=1\cdot2^{n-1} \end{gathered}[/tex]Now, we will compare the options:
The first term of both options is when you are four years old that n = 1
you only received money for ten years so, n = 10
So, substitute with n = 10 into both formulas:
[tex]\begin{gathered} Option1\rightarrow a_{10}=1000.50(10)=10005 \\ Option2\rightarrow a_{10}=1\cdot2^{10-1}=2^9=512 \end{gathered}[/tex]So, the answer will be:
For ten years, the option that gives the most money = Option 1
