Question Video: Solving Word Problems Involving Geometric Series Mathematics

The table below represents the salary of an employee in three consecutive years. The salary can be described by a geometric sequence. Find the total salary of the employee over 5 years.

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Video Transcript

The table below represents the salary of an employee in three consecutive years. The salary can be described by a geometric sequence. Find the total salary of the employee over five years.

We are told that in the first year, the employee earns 73,600 pounds, in the second year, they earn 110,400 pounds, and in the third year, they earn 165,600 pounds. We are told that this represents a geometric sequence. And in any geometric sequence, we have first term 𝑎 and common ratio 𝑟. This means that to get from the first term to the second term, we multiply by 𝑟. So if the first term is 𝑎, the second term is 𝑎𝑟. The third term is 𝑎 multiplied by 𝑟 multiplied by 𝑟, which is 𝑎𝑟 squared. This also means that the fourth term will be equal to 𝑎𝑟 cubed and the fifth term 𝑎𝑟 to the fourth power.

We can calculate the value of 𝑟 by dividing the second term by the first term, in this case, 110,400 divided by 73,600. This is equal to 1.5. Therefore, the common ratio of this geometric sequence is 1.5. We could use this to calculate the employee’s salary in the fourth and fifth years. 165,600 multiplied by 1.5 is equal to 248,400. This is the salary of the employee in pounds in the fourth year. Multiplying this by 1.5 gives us 372,600. In the fifth year, the employee earns 372,600 pounds. In order to find the total salary over five years, we could then sum these five values. This gives us a total salary of 970,600 pounds.

There is an alternative way of calculating the total after five years. We can do this by using the fact that in a geometric sequence, the sum of the first 𝑛-terms is equal to 𝑎 multiplied by 𝑟 to the power of 𝑛 minus one divided by 𝑟 minus one. This can also be written as 𝑎 multiplied by one minus 𝑟 to the power of 𝑛 divided by one minus 𝑟. In this question, as 𝑟 is greater than one, we will use the first formula. We know that 𝑎, the first term, is equal to 73,600 and 𝑟 is equal to 1.5. We need to calculate the total salary over five years. Therefore, 𝑛 is equal to five. 𝑠 sub five is equal to 73,600 multiplied by 1.5 to the power of five minus one all divided by 1.5 minus one.

The numerator simplifies to 485,300, and 1.5 minus one is equal to 0.5. As dividing by 0.5 is the same as multiplying by two, we once again get an answer of 970,600. This confirms that the total salary of the employee over five years is 970,600 pounds.

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