Respuesta :
Given that:
- The value of the computer when Robin bought it was $1,250.
- Its value will decrease by 10% each year that she owns it.
• You can identify that the first term of the sequence is:
[tex]a_1=1250[/tex]Knowing that it will depreciate by 10% each year, you can set up that the second term is (remember that a percent must be divided by 100 in order to write it in decimal form):
[tex]a_2=1250-(1250)(\frac{10}{100})[/tex][tex]\begin{gathered} a_2=1250-(1250)(0.1) \\ \\ a_2=1250-125 \\ \\ a_2=1125 \end{gathered}[/tex]Notice that the third term is:
[tex]\begin{gathered} a_3=a_2-(a_2)(0.1) \\ \\ a_3=1125-(1125)(0.1) \end{gathered}[/tex][tex]a_3=1012.5[/tex]And the fourth term:
[tex]\begin{gathered} a_4=a_3-(a_3)(0.1) \\ \\ a_4=1012.5-(1012.5)(0.1) \\ \\ a_4=911.25 \end{gathered}[/tex]By definition, a term of a Geometric Sequence is obtained by multiplying the previous term by a constant called Common Ratio.
Therefore, you can check it this sequence is geometric as follows by identifying if there is a Common Ratio:
[tex]\begin{gathered} \frac{a_4}{a_3}=\frac{911.25}{1012.5}=0.9 \\ \\ \frac{a_3}{a_2}=\frac{1012.5}{1125}=0.9 \\ \\ \frac{a_2}{a_1}=\frac{1125}{1250}=0.9 \end{gathered}[/tex]Notice that there is a Common Ratio:
[tex]r=0.9[/tex]Therefore, it is a Geometric Sequence.
• By definition, the Explicit Formula of a Geometric Sequence has this form:
[tex]a_n=a_1\cdot r^{n-1}[/tex]Where:
- The nth term is:
[tex]a_n[/tex]- The first term is:
[tex]a_1[/tex]- The Common Ratio is:
[tex]r[/tex]- The term position is:
[tex]n[/tex]In this case, you already know the value of the first term and the Common Ratio. Therefore, you can substitute them into the formula in order to represent the sequence:
[tex]a_n=(1250)(0.9)^{n-1}[/tex]• In order to find the value of the computer at the beginning of the 6th year, you need to find the sixth term of the sequence:
[tex]a_6[/tex]Therefore, you have to substitute this value of "n" into the Explicit Formula that represents the sequence and evaluate:
[tex]n=6[/tex]Then, you get:
[tex]\begin{gathered} a_6=(1250)(0.9)^{(6)-1} \\ a_6=(1250)(0.9)^5 \\ a_6=738.1125 \end{gathered}[/tex]Hence, the answers are:
• It is a Geometric Sequence because it has a Common Ratio.
,• Explicit Formula:
[tex]a_n=(1250)(0.9)^{n-1}[/tex]• Value of the computer at the beginning of the 6th year:
[tex]\text{ \$}738.1125[/tex]