The sequence formed by the value of the computer at the beginning of each year is geometric. This is because each year the value decreases by a constant factor of 10%.
The explicit formula for this geometric sequence is given by:
\[a_n = a_1 \times r^{(n-1)}\]
where:
- \(a_n\) is the value of the computer at the beginning of year \(n\)
- \(a_1\) is the initial value of the computer (\$1,250)
- \(r\) is the common ratio (1 - 0.10 = 0.90)
- \(n\) is the year number
We want to find the value of the computer at the beginning of the 6th year, so we plug in the values into the formula:
\[a_6 = \$1,250 \times 0.90^{(6-1)}\]
\[a_6 = \$1,250 \times 0.90^{5}\]
\[a_6 = \$1,250 \times 0.59049\]
\[a_6 = \$737.18625\]
Therefore, the value of the computer at the beginning of the 6th year will be approximately $737.19.
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.
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