1. Victor brought a brand new car for 750,000. IF the value of the car depreciates 20% per year, what will it be worth during the fourth year?
2. Given the terms a10 = 3/512 and a15 = 3/16384 of a geometric sequence, find the exact value of the first term of the sequence.
For 1, use the depreciation formula of p(1 - r)^t.
p = price of the brand new car ($750,000)
r = rate of 20% turned into 0.02
t = time of (4) years
Now, we substitute the values of the variables for the numbers involved.
750,000(1 - 0.02)^4 =
Next, you subtract 0.02 from 1, and then you raise that total to the fourth power.
1 - 0.02 = 0.98 = 0.98^4
0.98^4 raised to the fourth power equals 0.92236816.
Now, since you have 0.92236816 in the parenthesis, you multiply 750,000 by 0.92236816.
750,000(0.92236816) = ?
Also, don't forget that 750,000(0.92236816) is also the same as:
750,000 x 0.92236816 = ?
I hope this helps for 1! :)
P.S. Don't forget the formula that I used for depreciation.
To find the worth of Victor's car during the fourth year, we need to calculate the depreciation each year. Since the car depreciates by 20%, its value at the end of each year will be 80% (100% - 20%) of its previous year's value.
Here's the step-by-step process:
1. Start with the initial value of the car: $750,000.
2. Calculate the value at the end of the first year: $750,000 * 0.80 = $600,000.
3. Calculate the value at the end of the second year: $600,000 * 0.80 = $480,000.
4. Calculate the value at the end of the third year: $480,000 * 0.80 = $384,000.
5. Calculate the value at the end of the fourth year: $384,000 * 0.80 = $307,200.
Therefore, the car will be worth $307,200 during the fourth year.
Now let's move on to the second question:
To find the exact value of the first term of a geometric sequence, we can use the formula:
an = a1 * r^(n-1),
where 'a1' is the first term, 'r' is the common ratio, 'an' is the n-th term.
Given a10 = 3/512 and a15 = 3/16384, we can use these values to find the exact value of the first term.
Here's the step-by-step process:
1. Using a10 = 3/512, we can substitute the values in the formula: (3/512) = a1 * r^(10-1).
2. Simplifying this equation: (3/512) = a1 * r^9.
Repeat the same steps with the second equation, a15 = 3/16384: (3/16384) = a1 * r^(15-1) = a1 * r^14.
Now we have a system of two equations:
(3/512) = a1 * r^9,
(3/16384) = a1 * r^14.
To solve this system, we can divide the second equation by the first equation:
[(3/16384) / (3/512)] = [(a1 * r^14) / (a1 * r^9)].
Simplifying this equation:
[(1/16384) * (512/3)] = r^(14-9).
Now solve for 'r':
[1/2^14] * [2^9/3] = r^5,
1/3 = r^5,
r = ∛(1/3).
Finally, substitute this value of 'r' back into any of the original equations to find 'a1':
(3/512) = a1 * (∛(1/3))^9,
a1 = (3/512) * (∛(1/3))^(-9).
You can simplify this expression further using cube roots and exponentiation to find the exact value of the first term of the geometric sequence.