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Applications of Geometric Sequences and Series
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In this lesson, weโre going to see a lot of real-world problems which involve geometric sequences and series.
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Weโre going to apply everything we know about these to try and solve these real-world problems.
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Weโll see how to find the common ratio, how to find an explicit formula for the ๐th term, the order and value of the specific terms of the sequences, and how to find the sum of any number of given terms of our sequence.
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Before we start tackling real-world problems, letโs start by recalling everything we know about geometric sequences and series.
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First, we know that geometric sequences start with an initial value and we usually denote this ๐.
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Next, in a geometric sequence, we know thereโs always a common ratio between any two successive terms, and we usually call this ratio ๐.
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And equivalent to this, we can also say we multiply the previous term by ๐ to get the next term in our sequence.
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And on top of this, we could also have some formula to help us find information about geometric sequences and series.
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If we let ๐ข sub ๐ be the ๐th term in a geometric sequence with initial value ๐ and ratio of successive terms ๐, then we know ๐ข sub ๐ is equal to ๐ times ๐ to the power of ๐ minus one.
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Because we just need to multiply our initial value by ๐ ๐ minus one times to get the ๐th term in our sequence.
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Next, we also know we can find the ratio of successive terms in a geometric sequence by taking the quotient of two successive terms.
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๐ will be equal to ๐ข sub ๐ plus one divided by ๐ข sub ๐, and of course, this is provided that ๐ is greater than or equal to one.
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Weโve also seen how to take the sum of the first ๐ terms of a geometric sequence.
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We denote this ๐ sub ๐.
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Weโve shown that the sum of the first ๐ terms of this geometric sequence will be ๐ times one minus ๐ to the ๐th of power divided by one minus ๐.
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And there is a few things worth pointing out about this formula.
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First, itโs not valid when our value of ๐ is equal to one, because then weโre dividing by zero.
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But if ๐ was equal to one, our geometric sequence would just be ๐ over and over again.
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So, normally, we wouldnโt think about this as a geometric sequence anyway.
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Next, there is an equivalent formula we get by multiplying the numerator and denominator through by negative one, which is ๐ multiplied by ๐ to the ๐th power minus one all over ๐ minus one.
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And both of these will give the correct answer for any value of ๐ not equal to one.
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However, usually, we use the first version when our value of ๐ is less than one and the second version when ๐ is greater than one.
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However, it is personal preference which one you want to use.
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And finally, weโve also seen how to add an infinite number of terms of our geometric sequence together.
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We usually denote this ๐ sub โ.
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And we know itโs equal to ๐ divided by one minus ๐ if the absolute value of our ratio ๐ is less than one.
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And if the absolute value of ๐ is greater than or equal to one, our series will be divergent.
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Although there is one tiny caveat with this.
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We do need to check our value of ๐ is nonzero because then otherwise our sequence would just be zero over and over again.
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However, usually, we wouldnโt think of this as a geometric sequence anyway.
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Letโs now see some examples of how we can apply all of this to find information about real-world problems.
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Chloe joined a company with a starting salary of 28,000 dollars.
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She receives a 2.5 percent salary increase after each full year in the job.
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The total Chloe earns over ๐ years is a geometric series.
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What is the common ratio?
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Write a formula for ๐ sub ๐, the total amount in dollars Chloe earns in ๐ years at the company.
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After 20 years with the company, Chloe leaves.
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Use your formula to calculate the total amount she earned there.
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And thereโs one more part to this question we will discuss later.
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In this question, weโre given a real-world problem involving the amount of money Chloe will earn at a company over a period of time.
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Weโre told that Chloe has a starting salary of 28,000 dollars and that she receives a 2.5 percent salary increase after every full year she is in her job.
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In fact, this is enough information to determine that the amount she earns in ๐ years will be a geometric series.
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However, weโre also told this piece of information in the question.
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We need to determine the common ratio of this geometric series.
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To do this, we first need to recall that when we say the ratio of a geometric series, we mean the ratio of the geometric sequence which makes up this geometric series.
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And remember, in a geometric sequence, to get the next term in our sequence we need to multiply it by some common ratio we call ๐.
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We call ๐ the initial value of our sequence, and we call ๐ the common ratio.
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Then when we add these together because weโre adding terms of a geometric sequence together, we call this a geometric series.
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And remember, in this question, weโre told the total amount of money that Chloe earns in ๐ years at the company is the geometric series.
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So, each term in this series is actually going to be the amount of money Chloe earns each year.
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Thereโs a few different ways of finding this ratio.
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One way is to take the quotient of two successive terms.
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So, letโs find two of these successive terms.
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We can find the initial value by finding the amount of money Chloe will earn in year one in her company.
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And weโre told this in the question; itโs equal to 28,000 dollars.
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We then want to work out how much money she makes in the second year at the company.
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Remember, at this point, she will have worked one full year in her company, so she would have had a 2.5 percent salary increase.
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And thereโs a few different ways of calculating this value.
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For example, we could write this as 28,000 dollars plus 2.5 percent to 28,000 dollars.
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However, weโll write this as 28,000 dollars multiplied by 1.025.
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And this is enough to find the common ratio.
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However, there is one thing worth pointing out here.
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We can do exactly the same thing to find the amount earned in year three.
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Once again, sheโll get a 2.5 percent salary increase for working another full year, which means we would then need to increase the amount earned in year two by 2.5 percent.
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We would need to once again multiply this by 1.025.
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And this is, of course, true for any number of years.
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This is why this makes a geometric series.
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Now, thereโs a few different ways of finding our ratio ๐.
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For example, we could divide the amount made in year two by the amount made in year one.
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However, we can also notice weโre just multiplying by 1.025 each time.
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And thatโs enough to answer our question.
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The common ratio of this geometric sequence ๐ is going to be 1.025.
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The second part to this question wants us to writes a formula for ๐ sub ๐, the total amount of dollars Chloe earns in ๐ years at the company.
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Now, itโs worth pointing out we can just directly answer this question by using our formula for the sum of a geometric series.
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However, letโs first show why this is true.
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In this case, ๐ sub ๐ is the total amount in dollars that Chloe earns in ๐ years at the company.
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To find this value, we just need to add together the amount she earns in year one added to the amount she earns in year two, all the way up to the amount she would earn in year ๐.
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In the first year, weโve already shown she makes 28,000 dollars.
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In the second year, she gets a salary increase of 2.5 percent.
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So, sheโll make 28,000 multiplied by 1.025.
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Adding these two together gives the amount that she will earn in two years at the company.
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And, of course, we know this is true for any number of years, so we can keep going all the way up to the amount she will earn in year ๐.
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At this point in time, she will have worked ๐ minus one full years in the company.
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So, she would have got 2.5 percent salary increase ๐ minus one times.
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So, the amount she earns in year ๐ is 28,000 dollars multiplied by 1.025 raised to the power of ๐ minus one.
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And just as we showed before, this is a geometric series with initial value ๐, 28,000 dollars, and ratio of successive terms ๐, 1.025.
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And we know a formula to find the sum of the first ๐ terms of a geometric series.
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๐ sub ๐ will be equal to ๐ multiplied by ๐ to the ๐th power minus one all over ๐ minus one.
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So, we substitute ๐ is 28,000 dollars and ๐ is 1.025 into this formula to get that ๐ sub ๐ is equal to 28,000 dollars multiplied by 1.025 to the ๐th power minus one all over 1.025 minus one.
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And all we need to do is evaluate this expression, in our denominator, we have 1.025 minus one, which we can evaluate is 0.025.
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Then all we need to do is divide 28,000 by 0.025.
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And if we calculate this, we get 1,120,000, which gives us our final answer.
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๐ sub ๐, the total amount of dollars Chloe will earn in ๐ years at the company, is equal to 1,120,000 multiplied by 1.025 to the ๐th power minus one dollars.
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The third part of this question wants us to determine how much money Chloe will make if she leaves her company after 20 years.
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And weโre told to do this by using our formula.
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This is because we could just calculate the amount she earns in each year and then add all of these together.
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However, itโs far easier to use our formula for ๐ sub ๐.
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Remember, since weโre finding the amount she earns after 20 years at the company, our value of ๐ is going to be 20.
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So, we substitute ๐ is equal to 20 into our formula for ๐ sub ๐ we found in the previous question.
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We get ๐ sub 20 is equal to 1,120,000 multiplied by 1.025 to the 20th power minus one dollars.
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And if we just calculate this expression and give our answer to the nearest cent, we get 715,250 dollars 41 cents.
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But thereโs still one more part to this question to answer, so letโs clear some space.
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The last part of this question asked us to explain why the amount she earned will be different from the amount calculated using the formula.
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Option (A) she spent part of the money in 20 years.
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Option (B) the value of the dollar varies with time.
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Option (C) the actual amount will have a different percentage compared to the amount calculated using the formula.
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Option (D) the actual amount will have a different starting value compared to the amount calculated using the formula.
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Or option (E) when necessary, the new annual salary will be rounded.
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The last part of this question gives us an interesting problem.
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If Chloe were to calculate the amount she shouldโve earned by using our formula, she would find that her answer will be different from the actual amount she earned.
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Weโre given five possible options as to why this will be the case.
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We can actually answer this directly from our line of working.
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However, letโs just go through our five options first.
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Option (A) tells us that she will have spent part of the money in 20 years.
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Now, while it is true, she probably did spend part of the money in the 20 years, this will not affect the total amount that she earned in those 20 years.
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All this would really affect is the total amount of money she has left.
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So, option (A) is not the correct answer.
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Option (B) tells us the correct answer should be that the total amount she earned in 20 years will be different because the value of the dollar varies with time.
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And, of course, we do know it is true that the value of the dollar will vary with time.
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However, for the entire 20 years that Chloe worked for the company, she was paid in dollars.
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So, then no point would the value of the dollar change the total amount of money she made because she was only ever paid in dollars anyway.
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So, option (B) is not true.
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It wonโt change the total amount of money that she earned.
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However, you could make an argument that it would change the value of the amount of money that she made, but not the total.
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Option (C) tells us that we should have used a different percentage when we were calculating by using the formula.
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Once again, we know this wonโt be true because weโre told every year her salary will increase by 2.5 percent, and we used this value throughout.
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So, option (C) canโt be correct because we know her salary increases by 2.5 percent each year.
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Option (D) tells us that we should have used a different starting value for our formula.
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And once again, we know this is not true because weโre told in the question that her initial starting salary is 28,000 dollars.
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So, after one full year in her job, she will make 28,000 dollars.
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This will be the initial starting value.
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So, option (D) also canโt be correct.
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This then leaves us with option (E), which tells us, when necessary, the new annual salary will be rounded.
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Letโs discuss why this might change the actual amount that she earned.
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And to really highlight this, letโs calculate the amount of money she would earn in each year at her company.
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Letโs start with the first year.
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Of course, in the first year, she earns her starting salary 28,000 dollars.
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In the second year, sheโll earn the 28,000 dollars plus her salary increase of 2.5 percent.
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So thatโs 28,000 multiplied by 1.025.
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And if we calculate this value, weโll get 28,700 dollars exactly.
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And letโs do the same for year three and year four.
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In year three, weโll calculate that she makes 28,000 multiplied by 1.025 squared and in year four 28,000 dollars multiplied by 1.025 cubed.
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And if we calculate these, in year three, her salary is 29,417 dollars 50 cents.
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And in year four, we get 30,152 dollars and 93 cents.
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But we also get an extra 0.75 cents.
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And this is where our problem would start to rise because the company canโt give her 0.75 of a cent.
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So, most likely, the company would round up and give her 30,152 dollars and 94 cents.
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However, our formula adds together the exact amount calculated each year, whereas the actual amount she earned would be using the exact value she gets given.
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And this rounding can make our formula incorrect in this case.
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And itโs worth pointing out that this is only true when weโre working in dollars and cents because we canโt in this case give 0.75 of a cent.
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But this isnโt always true.
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For example, if we were working in length, then we can keep going as low as we want.
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This is why when working with real-world problems, itโs very important to know all of the things youโre working with.
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We were able to show that the amount that she earned was different to the amount we calculated using the formula because of options (E), when necessary, the new annual salary will need to be rounded.
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Letโs now see a different example of a real-world problem involving geometric sequences in series.
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A gold mine produced 2,257 kilograms in the first year but decreased 14 percent annually.
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Find the total amount of gold produced in the third year and the total across all three years.
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Give the answers to the nearest kilogram.
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In this question, weโre given some information about a gold mine.
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Weโre told that in the first year of production, the gold mine produces 2,257 kilograms.
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But weโre told that year on year, this amount is decreasing by 14 percent.
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The question wants us to find two things.
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It wants us to find the amount of gold which is produced in the third year of production, and it wants us to find the total amount produced in all three of the first years.
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And we need to give both of our answers to the nearest kilogram.
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Thereโs actually two different ways we can answer this question.
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The first way is to directly find these values from the information given to us in the question.
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Weโre told in the question, in the first year, the gold mine produces 2,257 kilograms of gold.
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We can find the amount of gold produced in the second year by remembering that this amount is going to decrease by 14 percent every year.
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Thereโs a few different ways of evaluating a decrease of 14 percent.
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One way is to multiply by one minus 0.14.
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And itโs worth pointing out here weโre subtracting 0.14 because this is a decrease.
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So we need to subtract, and we get 0.14 because our rate, ๐, is 14 and we need to divide this by 100.
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What weโre really saying here is a decrease in 14 percent is the same as multiplying by 0.86.
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Therefore, the amount of gold produced in the mine in the second year is 2,257 multiplied by 0.86 kilograms.
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And we can evaluate this exactly; we get 1941.02 kilograms.
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And we shouldnโt round our answer until the very end of the question, so weโll leave this in exact form.
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Weโre then going to want to do exactly the same for the third year.
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Once again, from the question, we know that the mine is going to produce 14 percent less gold in the third year than it did in the second year.
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So, one thing we could do is multiply the amount of gold we got in the second year by 0.86.
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However, itโs actually easy to just multiply our expression by 0.86.
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Multiplying this expression by 0.86 and simplifying, we get 2,257 multiplied by 0.86 squared kilograms.
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Calculating this expression exactly, we get 1669.2772 kilograms.
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We can now use these three values to answer our question.
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First, we can find the amount of gold produced in the third year by rounding this number to the nearest kilogram.
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This would then give us 1,669 kilograms.
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Next, we can find the total amount of gold produced in three years by adding these three values together.
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This gives us 2,257 kilograms plus 1941.02 kilograms plus 1669.2772 kilograms.
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And if we evaluate this expression, we get 5867.2972 kilograms.
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And to the nearest kilogram, we can see our first decimal place is two, so we need to round down, giving us 5,867 kilograms.
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However, what would have happened if we needed to find even more years of production?
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We can see that this method only really worked because we only had to calculate the first three years.
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If we were asked to find even more years in our example, we would need to notice something interesting.
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Each year weโre multiplying by a constant ratio of 0.86.
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And remember, in a sequence, if weโre multiplying by a constant ratio to get the next term in our sequence, we call this a geometric sequence.
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So, the gold produced in our mine after ๐ years forms a geometric sequence with initial value ๐, 2,257 kilograms, and ratio ๐, 0.86.
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We can then use what we know about geometric sequences to find the amount of gold produced after ๐ years in our mine and the total amount of gold produced after ๐ years.
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We just substitute ๐ is equal to three and our values for ๐ and ๐ into the two formula to find these expressions.
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And after rounding, we get the same answers we had before.
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๐ sub three will be 1,669 kilograms and ๐ sub three will be 5,867 kilograms.
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Letโs now go over the key points of this video.
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First, a lot of real-world problems involve geometric sequences and series.
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We also know if we can turn any of these real-world problems into a problem involving geometric sequence or series, then we can use any of the results we know about geometric sequences or series to help us answer these questions.
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Finally, we should always check that our answers make sense in the real-world situation weโre given.
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For example, sometimes, our calculations will involve noninteger values for populations.
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And we always need to be worried how this might affect our final answer.