Get ready for AP® Calculus
- Geometric series introduction
- Finite geometric series formula
- Worked examples: finite geometric series
- Geometric series formula
- Geometric series word problems: swing
- Geometric series word problems: hike
- Finite geometric series word problems
- Geometric series with sigma notation
- Worked example: finite geometric series (sigma notation)
- Finite geometric series
Modeling the length of a hike with a geometric series.
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- Throughout the video Sal is 1.2 in place of 20%. But 20% is equivalent to 0.2. How come he is using 1.2 instead of 0.2 if that's the equivalent?(26 votes)
- Sal is using 1.2 because the hiker is hiking 20% MORE than the previous day. If the common ratio was 0.2, then the hiker would be hiking less than the previous day because the value is >1. Therefore, the common ratio is 1.2.(49 votes)
Is there another formula to solve this equation? Like a much simpler one? If not, can you explain this one in detail? Thanks! :D(4 votes)
- there are 2 variations of the same formula:
multiply numerator and denominator by -1
here's proof for the formula
Sₙ = a + ar + ar² + ... + arⁿ⁻² +arⁿ⁻¹
→ the reason why the last term is not arⁿ:
S₃ = a + ar + ar²
you see the last term's power is 3-2
multiply A by r
rSₙ = ar + ar² + ... + arⁿ⁻² + arⁿ⁻¹ + arⁿ
A and B has the same amount of terms shown (5)
so if we do B - A:
rSₙ - Sₙ = -a + ar - ar + ar² - ar² + arⁿ⁻² - arⁿ⁻² + arⁿ⁻¹ - arⁿ⁻¹ + arⁿ
rSₙ - Sₙ = arⁿ - a
Sₙ(r - 1) = a(rⁿ - 1)
finally we end up with:
Sₙ = a(rⁿ - 1)/(r - 1)
hope it helped you to understand the formula... sorry for the 2 year late reply(2 votes)
- How can Sal switch around 1-1.2^4 with 1.2^4-1? And why does he switch them? It would be changing it from negative to positive wouldn't it?(3 votes)
- he can switch it because the denominator is also negative, and the numerator is negative too. you switch them, and then both are now positive.
just like (1-2)/(1-3)=(2-1)/(3-1) because -1/-2= 1/2.(4 votes)
- Following this example, we add the 20% to the first day to find that r=1.2. However, in the practice part of this section there is this question
Vince went on a 3 day hiking trip. Each day, he walked 3/4 the distance that he walked the day before. He walked 83.25 kilometers total in the trip. How far did he walk on his first day?
Thus r would equal 1.75. However, the hints shows how the solution is found and in those hints it states that r=.75 and no 1.75. Where is the mistake? Who is correct? Who should I listen to? Please explain.
I came up with 83.25 = x(1-(1.75^3))/(1-1.75) where x came out to be just over 14km(2 votes)
- The difference between the example and the practice problem is in the question itself. In the video the difference is increasing by 20%, making 1.2 correct. However, if you were to walk 20% of the distance as the day before, that would mean it is decreasing and you would then use 0.2. For the practice problem, the hint is correct and you would use 0.75 since multiplying by 0.75, or in fraction form 3/4, gives you three fourths of the original value. Similarly, multiplying by 1.2 gives you a value that is twenty percent larger than the original, while multiplying 0.2 would give you a value that is twenty percent of (or otherwise put, 80 percent less than) the original.(4 votes)
- A concept I've always struggled with concerning variables is how are you able to separate the a from the fraction in the hiking word problem?
What I mean is, for
a(1.2^4-1) / 0.2 = 27
You were able to simplify it to
a(1.2^4-1 / 0.2) = 27
if you were to separate the terms, why would it not be
a / 0.2 * (1.2^4-1) / 0.2 = 27
Is the 0.2 not dividing both the a term and the term within the parenthesis?
The way I'm thinking is, if the coefficient of a is 1, it would be
1a / 0.2 * (1.2^4-1) / 0.2(2 votes)
- Well, consider it like this. We're simply pulling a factor out of our full equation; e.g. let's say that instead of a(1.2^4-1)/0.2 = 27 we had (4 * 8)/2 = 16.
We can either represent this as:
(4 * 8)/2 = 16
Or we can represent it as:
4 * (8/2) = 4 * (4) = 16
If it helps, note that (4*8)/2 is an equivalent to (4/1) * (8/1) * (1/2). On a similar note, a(1.2^4-1)/0.2 is the same as (a/1) * (1.2^4-1)/1 * (1/0.2). It's a matter of how you represent the equation, with the importance being that you maintain equivalency in your simplifications. I hope this is helpful!(3 votes)
- Why isn't sal using the 27=[a(1-(1.2)^4)]/1-(1.2) but instead, he is using 27=[a-a(1-(1.2)^4)]/1-(1.2)?(2 votes)
- shouldn't the rate be to the power of 3 because its the fourth term, so it would be n-1, which is 4-1=3?(2 votes)
- The sum of the first 𝑛 terms of a geometric sequence is
𝑎(1 − 𝑟^𝑛)∕(1 − 𝑟)
However, the 𝑛-th term of a geometric sequence is
𝑎𝑟^(𝑛 − 1)(2 votes)
- If a= the first term
does this mean that the sum of n terms is equal a(1.2)^n-1 ?
And if so, why can't you solve the problem in the video by following these steps:
- At3:02, it is said that in other videos, where the formula for the sum of first n terms comes from is explained. Can someone either:
(1) tell me which video this is
(2) explain to me where the formula for the sum of the first n terms is derived from(2 votes)
- [Instructor] We're told Sloan went on a four-day hiking trip. Each day, she walked 20% more than the distance that she walked the day before. She walked a total of 27 kilometers. What is the distance Sloan walked in the first day of the trip? And it says to round our final answer to the nearest kilometer. So, like always, have a go with this and see if you can figure out how much she walked on the first day. All right, well let's just call the amount that she walked on the first day a. And then using a, let's see if we can set up an expression for how much she walked in total. And then, that should be equal to 27. And then hopefully, we're going to be able to solve for a. So the first day, she walks a kilometers. Now, how about the second day? Well, they tell us that each day, she walked 20% more than the distance she walked the day before. So, on the next day, she's going to walk 20% more than a kilometers, so that's 1.2 times a. And what about the day after that, her third day? Well, that's just gonna be 1.2 times this, the second day, and so that's going to be 1.2 times 1.2, or we can say 1.2 squared times a. And then how much on the fourth day? And that's, she went on a four-day hiking trip, so that's the last day. Well, that's gonna be 1.2 times the third day. So, that's going to be 1.2 to the third power times a. So, this is an expression in a on how much she walked over the four days. And we know that she walked a total of 27 kilometers, so this is going to be equal to 27 kilometers. Now, you could solve for a over here. You could factor out the a and you could say a times one plus 1.2 plus 1.2 squared plus 1.2 to the third power is equal to 27. And then you could say that a is equal to 27 over one plus 1.2 plus 1.2 squared plus 1.2 to the third power, and we would need a calculator to evaluate this. But I'm gonna do a different technique, a technique that would work even if you had 20 terms here. You, in theory, could also do this with 20 terms but it gets a lot more complicated, or if you had 200 terms. So the other way to approach this is use the formula for a finite geometric series. What does it evaluate to? And just as a reminder, the sum of first n terms, it's going to be the first term, which we could call a, minus the first term times our common ratio. In this case, our common ratio is 1.2 because every successive term is 1.2 times the first. So our first term times our common ratio to the nth power, all of that over one minus the common ratio. In other videos, we explain where this comes from, we prove this, but here, we can just apply it. We already know what our a is, I used that as our variable. Our common ratio in this situation is going to be equal to 1.2. And our n is going to be equal to four. Another way I like to think about it is it's our first term, which we see right over there, minus the term that we did not get to. If we were to have a fifth term, it would have been that fifth term that we're subtracting because we aren't getting to a fourth power here, the fifth term would have been the fourth power, all of that over one minus the common ratio. And so, this left-hand side of our equation, we could rewrite as our first term minus our first term times our common ratio, 1.2, to the fourth power. All of that over one minus our common ratio. And then that could be equal to 27. Let me scroll down a little bit so we have some more space to then solve this. And so, let's see, I can simplify this a little bit. This is going to be equal to negative 0.2. Our numerator, we can factor out a. And so, this is going to be equal to a times one minus 1.2 to the fourth power. And let's see, we can multiply both the numerator and the denominator by negative one. And so, this would get us to a times, a times, and I'll put the a out of the fraction, a time, so I'll just swap the order here and get rid of this negative. 1.2 to the fourth power minus one over 0.2 is equal to 27. Again, all I did is I took the a out of the fraction so it's out here, and I multiplied the numerator and the denominator by negative. The numerator multiplied by negative would swap these two. And then multiplying negative 0.2 times negative is just positive 0.2. And so now, I can just multiply both sides times the reciprocal of this. So, I'll do up here. So, 0.2 over 1.2 to the fourth minus one. And then here, 0.2 over 1.2 to the fourth minus one. That cancels with that, that cancels with that, that's exactly why I did that. And we're left with a is equal to, it is equal to, I'll just write it in yellow, 27 times 0.2, all of that over 1.2 to the fourth minus one. And this expression should give us the exact same value as that expression we just saw, but this is useful even if we had a hundred terms, we could use this. And so, I'll get the calculator out. This will give us... So, actually, I'll evaluate this denominator first, so I'll have 1.2 to the fourth power, which is equal to minus one, is equal to, and that's in the denominator, so I could just take the reciprocal of it, and then multiply that, times 27 times 0.2 is equal to 5.029. Now, they want us to round our answer to the nearest kilometer, so this is going to be approximately equal to, approximately equal to five kilometers. That's how much approximately that she would have traveled on the first day of her hiking trip.