Video: Solving Word Problems Involving Geometric Sequences

The monthly salary of an employee this year is 2,100 LE. Find his salary after 8 years given he received a 3% raise every year, giving the answer to the nearest pound.

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Video Transcript

The monthly salary of an employee this year is 2,100 Egyptian pounds. Find his salary after eight years given he received a three percent raise every year, giving the answer to the nearest pound.

In this question, we have some information about the starting monthly salary of an employee. We’re also told that he’s given a three-percent raise every year, per annum. And we’re looking to find out what his salary is eight years after the start. Now, let’s go back to the information about the three-percent raise. We know if his starting salary is 100 percent, then a three-percent raise is finding 100 plus three percent or 103 percent of the starting salary. We can divide this by 100 to work out what 103 percent is as a decimal. When we do, we get 1.03. Now, this is called a decimal multiplier. It’s the single value that we multiply the original salary by to calculate the salary in the following year.

Notice that this value remains unchanged. And so, in fact, if we look carefully, we see that we can in fact describe this as a geometric sequence. A geometric sequence or a geometric progression is one where each term is found by multiplying the previous one by a fixed nonzero number, called the common ratio. The 𝑛th term of a geometric progression is 𝑎 times 𝑟 to the power of 𝑛 minus one, where 𝑎 is the first term in the sequence and 𝑟 is the common ratio.

Now, the first term in our sequence is the starting salary of our employee. It’s 2,100 or 2,100 Egyptian pounds. We saw that to find the new salary each year, we multiply the previous one by 1.03. So the common ratio is 1.03. And we’re looking to find the salary after eight years. So we’re looking to find the eighth term in our sequence, so we let 𝑛 be equal to eight. This means the eighth term, 𝑎 sub eight, is 2,100 times 1.03 to the power of eight minus one or to the seventh power. That gives us 2,582.735 and so on. Correct to the nearest whole number, and therefore the nearest pound, that’s 2,583.

And so we see that the monthly salary of this employee after eight years is 2,583 Egyptian pounds.

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