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Geometric series word problems: swing

CCSS.Math:

Video transcript

we're told a monkey is swinging from a tree on the first swing she passes through an arc of 24 meters with each swing is she passes through an arc 1/2 the length of the previous swing so what's going on here let's say this is the top of the rope or the vine that the monkey is swinging from and so on that first swing I could draw a little monkey here so this is my little monkey so on the first swing the monkey will go 24 meters so might do something like this then that arc is 24 meters and then on the second swing it would be she'd swing back it an arc half the length of the previous swing so then she would come back and then it would be half the length and so maybe swing back over here and then on the next so that would be 12 and then on the next swing she would swing half of that which would be 6 meters and so she might swing like this and that makes sense that's consistent with our experiences swinging from trees for those of us who have done that so let's look at the first choice which expression gives a total length the monk the monkey swings in her first n swings so pause the video and see if you can do that and you can express it as actually express it about two ways express it as a geometric series but also express it as the sum of a geometric series if we were actually evaluated so let's do this together so we already said on the first swing the monkey goes 24 meters now on the second swing and I gave you a hint when I said to express it as a geometric series she swings half that now I could just write a 12 here but the half is interesting because that's going to be my common ratio for my geometric series every successive swing the arc length is half the arc length of the last swing so it's going to be it's going to be 24 times 1/2 and then on the next swing it's going to be 24 it's going to be 1/2 of this so it's going to be 24 times 1/2 times 1/2 so that's 24 times 1/2 to the second power and so this would be the first three swings notice that the exponent here we got to the second power so the first ends we are going to get 224 times 1/2 not to the enth power but to the N minus 1 power notice after 2 swings we only get 224 times 1/2 to the first power after 30 swings to the second power so after and swings to the n minus 1 power now as I said we don't want to just have this expression we actually want to know what how do we evaluate this and the way we evaluate this is we look at the formula which we've explained and we've proven in other videos the formula for a finite geometric series so that tells us and I'll just write it over here the sum of first first n terms is a where a is the first term so that's going to be our 24 in this situation it's a minus a times our common ratio already said that our common ratio is 1/2 to the nth power so one way I like to remember it is it is our first term minus the first term that we didn't include or minus what would have been the term right after this all of that over 1 minus our common ratio and there's other ways that you might have seen this written you could factor an a out and you might have seen something like this a times 1 minus R to the N all of that over 1 minus R these two are equivalent but now let's use this so this is going to be equal to actually I'll use this second form right over here so our first term a is 24 so we're going to have 24 times 1 minus our common ratio which is 1/2 to the nth power well we're talking about the first n swing so I'm just going to leave an N right over there all of that over 1 minus our common ratio 1 minus 1/2 so we could leave it like that or we could simplify it a little bit if we like 1 minus 1/2 is equal to 1/2 24 divided by 1/2 is equal to 48 so if you wanted to you could simplify it to 48 times 1 minus one half to the nth power so either of these would be legitimate now the second part they say what is the total distance the monkey has traveled when she completes her twenty-fifth swing and they say round your final answer the nearest meter so pause this video and see if you can work that out all right well we can just use this expression here and we know that we are completing our twenty-fifth swing so n is 25 and so we'll just put a 25 there so that's going to be 48 times 1 minus 1/2 to the 25th power now this is going to be a very very small very very small number so it's actually going to be pretty close to 48 meters but let's see what this is equal to and we're going to round to the nearest meter all right so let's get our calculator out and so let's just evaluate one half I'll just write that is 0.5 to the 25th power which as we said as we predicted is a very small number and then we're going to subtract that from one so it's put a negative and then I'll add one to it and so that is very close to one and so my prediction is holding true so I multiply that times 48 if we round to the nearest meter we get back to 48 meters so this is going to be 48 meters and we're done