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.
An alternative way to calculate the value of the computer at the beginning of the 6th year is to use the formula for the nth term of a geometric sequence:
\[ a_n = ar^{n-1} \]
where:
- \( a = 1250 \) (initial value of the computer)
- \( r = 0.9 \) (common ratio)
- \( n = 6 \) (6th year)
Plugging in these values, we get:
\[ a_6 = 1250 \times 0.9^{6-1} \]
\[ a_6 = 1250 \times 0.9^5 \]
\[ a_6 = 1250 \times 0.59049 \]
\[ a _6 = 738.1125 \]
So, the value of the computer at the beginning of the 6th year is $738.11.
Are you sure?
Let's double-check the calculation to ensure accuracy.
\[ a_6 = 1250 \times 0.9^{5} \]
\[ a_6 = 1250 \times 0.59049 \]
\[ a_6 = 738.1125 \]
Therefore, the correct value of the computer at the beginning of the 6th year is indeed $738.11.