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## Algebra 2

### Course: Algebra 2>Unit 3

Lesson 7: Geometric series

# Geometric series word problems: hike

Can a finite geometric series help us plan a hiking trip? Sloan embarks on a four-day hiking trip, increasing her distance by 20% each day. To figure out her first day's distance, we use the finite geometric series formula. This math tool helps us solve Sloan's hiking puzzle.

## Want to join the conversation?

• 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?
• 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.
• QUESTION

Is there another formula to solve this equation? Like a much simpler one? If not, can you explain this one in detail? Thanks! :D
• there are 2 variations of the same formula:
`Sₙ=a(1-rⁿ)/(1-r)`
multiply numerator and denominator by -1
`Sₙ=a(rⁿ-1)/(r-1)`

here's proof for the formula
A:`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
B:`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`
simplify:
`rSₙ - Sₙ = arⁿ - a`
factorize completely:
`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
• 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
• 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.
• 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?
• 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.
• At , how is 20% more each week equal to 1.2? When I paused the video to solve for a, I wrote 1/5 instead of 1.2, is that correct?
• 20% of something is 1/5, yes. So 20% more is the original amount (1 times the original) plus 1/5 the original, or 1+(1/5)=1.2 times the original.
• Hello everyone!

My question is not exactly on topic, but I wanted know: Is there a method that can be used to solve any polynomial?

I already think there isn't but I really want to be double checked sure.

Thanks ^^
• No, not every polynomial has solutions in terms of addition, subtraction, multiplication, division, and roots. Every polynomial of degree 4 and less does, but not higher-degree polynomials.
• 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
• 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!
• Why in some of the questions you translate 20% as .20 but in others you translate it to 1.20?
• Can’t we just simplify the formula as a(r^n-1)/r-1. If not, why?