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Robin bought a computer for $1,250. It will depreciate, or decrease in value, by 10% each year that she owns it.
Is the sequence formed by the value at the beginning of each year arithmetic, geometric, or neither? Explain.
Write an explicit formula to represent the sequence.
Find the value of the computer at the beginning of the 6th year.
The sequence formed by the value of the computer at the beginning of each year is geometric. This is because the value decreases by the same percentage each year.
Let's denote the initial value of the computer as $1250. Since the computer depreciates by 10% each year, the common ratio (r) is 0.9 (100% - 10% = 90% = 0.9).
The explicit formula to represent the sequence is:
\[ a_n = 1250 \times (0.9)^{n-1} \]
Where:
- \(a_n\) is the value of the computer at the beginning of year n.
To find the value of the computer at the beginning of the 6th year (n=6), we substitute n=6 into the formula:
\[ a_6 = 1250 \times (0.9)^{6-1} \]
\[ a_6 = 1250 \times 0.9^5 \]
\[ a_6 = 1250 \times 0.59049 \]
\[ a_6 = 738.1125 \]
Therefore, the value of the computer at the beginning of the 6th year is $738.11.
Are you sure?
Let's verify the calculation for the value of the computer at the beginning of the 6th year.
Given:
Initial value of the computer, \(a_1 = $1250\)
Common ratio, \(r = 0.9\)
Year we want to find the value for, \(n = 6\)
The explicit formula for the sequence is:
\[ a_n = 1250 \times (0.9)^{n-1} \]
Substitute the values for n into the formula:
\[ a_6 = 1250 \times (0.9)^{6-1} \]
\[ a_6 = 1250 \times 0.9^5 \]
\[ a_6 = 1250 \times 0.59049 \]
\[ a_6 = 738.1125 \]
Therefore, the value of the computer at the beginning of the 6th year is indeed $738.11. Thank you for asking to recheck.