The depreciation factor follows the sequence below:
8%, 12%, 18%, 27%
The difference between consecutive values are +4, +6, +9
The second differences are +2, +3. They are not equal, so the sequence is not quadratic.
To find out if this is a geometric sequence, we divide consecutive terms as follows:
[tex]\begin{gathered} \frac{12}{8}=1.5 \\ \frac{18}{12}=1.5 \\ \frac{27}{18}=1.5 \end{gathered}[/tex]
Since all of these ratios are equal, this is a geometric sequence with a common ratio of r = 1.5 (or 3/2).
b)
The formula for the n-th term of a geometric sequence is:
[tex]a_n=a_1\cdot r^{n-1}[/tex]
Where a1 is the first term. Substituting:
[tex]a_n=8\cdot\mleft(\frac{3}{2}\mright)^{n-1}[/tex]
c) It can be helpful to the bike owner because if he/she owns a bike for more than 4 years, they could estimate the depreciation % even if it's not present in the table.
For example, if the length of the ownership is n = 6 years, the depreciation factor is:
[tex]\begin{gathered} a_6=8\cdot\mleft(\frac{3}{2}\mright)^{6-1} \\ a_6=8\cdot(1.5)^5 \\ a_6=60.75 \end{gathered}[/tex]
The bike is depreciated by 60.75%
2)
a)
The sum of the terms is:
8% + 12% + 18% + 27% + 40.5% + 60.75% + ...
The last two terms were obtained with the formula.
The sum of n=1 is S1 = 8
The sum of n=2 is S2 = 20
The sum of n=3 is S3 = 38
The sum of n=4 is S4 = 65
The sum of the n first terms of a geometric series is:
[tex]S_n=a_1\cdot\frac{r^n-1}{r-1}[/tex]
Substituting a1 = 8, r = 1.5:
[tex]\begin{gathered} S_n=8\cdot\frac{1.5^n-1}{1.5-1} \\ S_n=8\cdot\frac{1.5^n-1}{0.5} \\ S_n=16\cdot(1.5^n-1) \end{gathered}[/tex]
b) For the bike owner, the series represent the cumulative depreciation factor after n years of ownership.
c) I don't see why it could be helpful to the bike owner because the depreciation factor is not cumulative. He/she won't matter for the sum of the factors.
