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Question Video: Solving Word Problems by Forming Geometric Sequences Mathematics • Second Year of Secondary School

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The value of a brand-new car which costs 498,300 LE decreases at a rate of 15% every year. Find the value of the car after 8 years, giving the answer to the nearest pound.

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Video Transcript

The value of a brand-new car which costs 498,300 Egyptian pounds decreases at a rate of 15 percent every year. Find the value of the car after eight years, giving the answer to the nearest pound.

For this question, we’ve been given some information about the starting value of a car. We’re told that its value decreases by 15 percent every year. And we’re asked to find the value of the car after eight years.

Let’s begin with the information about the depreciation of the car; that’s the decrease in value over time. We’re told that the car loses 15 percent of its value every year. So, if we take the starting value to be 100 percent, then we’re going to subtract 15 percent from this. We find that the new value of the car each year is 85 percent of its value from the previous year. We can also divide this value by 100 to find the decimal equivalent of 85 percent is 0.85. In doing this, we have found that the value of the car in a given year can be found by multiplying its value from the previous year by 0.85. Note that the question also tells us that this happens every year.

With this information, we can form a pattern that allows us to represent the yearly values of the car as a geometric sequence. Remember, a geometric sequence, or a geometric progression, is one where each term is found by multiplying the previous term by a fixed nonzero number called the common ratio. The 𝑛th term in a geometric sequence can be represented as 𝑎 sub 𝑛. This 𝑛th term can be found by multiplying the first term, 𝑎, by the common ratio, 𝑟, raised to the power of 𝑛 minus one.

In this case, we define the first term, 𝑎, as the starting value of the car, which is 498,300 Egyptian pounds. We have previously discussed that each year the value of the car can be found by multiplying the value from the previous year by 0.85. Given that this multiplier is the same every year, this is our common ratio 𝑟.

Finally, we need to find the value of the car after eight years. This means that 𝑛 is equal to eight. And we’ll be finding 𝑎 sub eight, which is the eighth term in the sequence. We can find 𝑎 sub eight using the formula for the 𝑛th term of a geometric sequence and substituting in our values. This gives us that 𝑎 sub eight is equal to 498,300 multiplied by 0.85 to the power of eight minus one, in other words, 0.85 to the power of seven. This is equal to 159,743.563 and so on.

Remember, this question asks us to give our answer to the nearest pound. If we consider our units, this means we must round our answer to the nearest whole number, that is, 159,744. In doing this, we have found the answer to the question. To the nearest pound, the value of the car after eight years is 159,744 Egyptian pounds.

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