Robin bought a computer for $1,250. It will depreciate, or decrease in value, by 10% each year that she owns it.
a) Is the sequence formed by the value at the beginning of each year arithmetic, geometric, or neither? Explain.
b) Write an explicit formula to represent the sequence.
c) Find the value of the computer at the beginning of the 6th year.
yes
a) The sequence formed by the value at the beginning of each year is geometric. This is because the value of the computer decreases by a constant factor of 10% each year.
b) To write an explicit formula for the sequence, we can use the formula for a geometric sequence:
\[a_n = a_1 \cdot r^{n-1}\]
where \(a_n\) represents the value of the computer at the beginning of the nth year, \(a_1\) is the initial value of the computer, and \(r\) is the common ratio representing the decrease in value each year.
In this case, the initial value is $1,250 and the common ratio is 1 - 0.10 = 0.90 (or 90% of the previous year's value). Therefore, the explicit formula for the sequence is:
\[a_n = 1250 \cdot 0.90^{n-1}\]
c) To find the value of the computer at the beginning of the 6th year, we substitute n = 6 into the explicit formula:
\[a_6 = 1250 \cdot 0.90^{6-1}\]
Simplifying, we get:
\[a_6 = 1250 \cdot 0.90^5\]
Calculating the value, we find:
\[a_6 \approx 913.13\]
Therefore, the value of the computer at the beginning of the 6th year is approximately $913.13.
a) To determine whether the sequence formed by the value of the computer at the beginning of each year is arithmetic, geometric, or neither, we need to see if there is a constant ratio or difference between the terms.
Since the computer's value depreciates by 10% each year, we can say that the value at the beginning of each year is multiplied by 90% (100% - 10%). Therefore, there is a constant ratio between the terms, making the sequence geometric.
b) To write an explicit formula to represent the sequence, we can start with the initial value of the computer and then multiply it by the common ratio each year.
Let's represent the value of the computer at the beginning of the year as V and the year number as n. Since the computer depreciates by 10% each year, the common ratio is 90% or 0.9.
So the explicit formula for the sequence would be:
V(n) = V * (0.9)^(n-1)
c) To find the value of the computer at the beginning of the 6th year, we can substitute n = 6 into the explicit formula:
V(6) = V * (0.9)^(6-1)
V(6) = V * (0.9)^5
Now, let's calculate the value of the computer at the beginning of the 6th year by substituting the given information:
V = $1,250
n = 6
V(6) = $1,250 * (0.9)^5
V(6) ≈ $1,250 * 0.59049
V(6) ≈ $737.61
Therefore, the value of the computer at the beginning of the 6th year is approximately $737.61.
Value = 1250(.9)^n
does that look arithmetic or geometric to you ?