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Current time:0:00Total duration:6:30

Geometric series word problems: hike

CCSS.Math:

Video transcript

we're told Sloane went on a four-day hiking trip each day she walked 20 percent more than the distance that she walked the day before she walked a total of 27 kilometers what is the distance Sloane walked in the first day of the trip insist around our final answer to the nearest kilometer so like always have a go at 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 in the first day she walks a kilometers now how about the second day well they tell us that each day she walked 20 percent more than the distance she walked the day before so on the next day she's going to walk 20 percent 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 going to be 1.2 times this the second day and so that's going to be one point two times 1.2 or we could 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 going to be 1.2 times the third day so that's going to be one point two 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 1 plus 1 point 2 plus 1 point 2 squared plus 1 point 2 to the third power is equal to 27 and then you could say that a is equal to 27 over 1 plus 1 point 2 plus 1 point 2 squared plus 1 point 2 to the third power and we would need a calculator to evaluate this but I'm going to do a different technique a technique that would work even if you had 20 terms here you in theory could also do this with twenty terms but it gets a lot more complicated or if you had two hundred 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 1 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 use 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 4 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 to the fourth power all of that over 1 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 1 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 we could this is going to be equal to negative 0.2 our numerator we can factor out an A and so this is going to be equal to a times 1 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 I'll put the a out of the out of the fraction a times so I'll just swap the order here and get rid of this negative one point two to the fourth power minus one over zero point two 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 in the denominator by a negative the numerator multiplied by negative would swap these two and then multiplying negative 0.2 times a negative is just positive zero point two and so now I can just multiply both sides times the reciprocal of this so I'll do it here so zero point two over one point two to the fourth minus one and then here zero point two over one point two 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 zero point two all of that over one point two to the fourth minus one and this expression should give us the exact same value is 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 I will actually I'll evaluate this denominator first so I'll have one point two to the fourth power which is equal to minus one is equal to now it'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 5.0 to nine 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