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Sequences word problems

Sal solves two word problems about modeling real-world situations with arithmetic and geometric sequences.

Video transcript

- [Voiceover] Mohamed decides to track the number of leaves on the tree in his backyard each year. The first year, there were 500 leaves. Each year thereafter, the number of leaves was 40% more than the year before. Let n be a positive integer, and let f of n denote the number of leaves on the tree in Mohamed's back yard in the nth year since he started tracking it. The expression f of n defines a sequence. What kind of sequence is f of n? So, some of you might be able to think about this in your head. Each successive year we're growing by 40%, that's the same thing as multiplying by 1.4. Each successive term we're multiplying or dividing by the same number. Well, that's going to be geometric. Let's make that a little bit more tangible, just in case. So, let's make a little table here. So, table. So, this is n and this is f of n. So when n is equal to one, the first year, n equals one, there were 500 leaves. F of n is 500. Now, when n is equal to two we're going to grow by 40%, which is the same thing as multiplying by 1.4. So 500 times 1.4, let's do 40% of 500 is 200, so we're going to grow by 200, so we're going to go to 700. Then in year three, we're going to grow by 40% of 700, which is 280, so it's going to grow to 980. Notice it's definitely not an arithmetic sequence. An arithmetic sequence, we would be adding or subtracting the same amount every time, but we're not. Here, from 500 to 700, we grew by 200, and then from 700 to 980, we grew by 280. Instead, we're multiplying or dividing by the same amount each time. In this case, we're multiplying by 1.4, by 1.4 each time. So we are clearly geometric. Depending on your answer to the question above, the recursive definition of the sequence can have one of the following two forms. Well this is the arithmetic form, which we know isn't the case, so it's going to be in the geometric form. And then they ask us, what are the values of the parameters A and B for the sequence? So we have our base case here, f of n is going to be equal to A when n is equal to one. Well, we know that when n equals one, we had 500 leaves on the tree so A, this A over here, is 500, so A is 500, and then if we're not in the base case for any other year, we are going to have, let's see, it's going to be the previous year, the previous year times what? It's going to be the previous year grown by 40%, to grow by 40%, you're going to multiply by 1.4 so B is going to be 1.4. You take the previous year and you multiply by 1.4 for any other year, any year other than n equals one. So, B is equal to 1.4, and we're done. Let's do another example. This is strangely fun. All right, so this says: Seo-Yun hosted a party. She had 50 party favors to give away, and she gave away three party favors to each of her guests as they arrived at the party. Let n be a positive integer, and let g of n denote the number of party favors Seo-Yun had before the nth guest arrived. All right, actually, before I even look at these questions, let me make a table here because they're saying before the nth guest. I want to make sure I'm understanding this properly. So this is n, and then this is going to be g of n, right over here. So, when n is equal to one, when n is equal to one, g of n is going to be, or g of one is going to be the number of party favors Seo-Yun had before the first guest. Well, before the first guest, she had 50 party favors. She had 50 party favors. Now, the second guest comes. Now the number of party favors she had before the second guest, well, she had to give three to the first guest, so she's now going to have 47 party favors. Now, when n is equal to three, how many party favors did she have before the third guest? Well, she would've had to give party favors to the first and second guest, who each got three. So, she would have 44, and I think you see the pattern. For every time n, when n equals one, g of n is 50, and every time we increase n by one, every time we increment n, we are increasing g of n by plus three, by minus three, I should say because she's giving away party favors, minus three. Minus three. So because the difference between successive terms is the same, we know this is an arithmetic sequence. This is an arithmetic sequence and then they say write an explicit formula for the sequence. So let's think about this. Let's see, g of n is going to be equal to... Let's see, we're going to start at 50 then we're going to subtract three, and let's think, do we subtract three times n or is it...? Let's see, for the first guest, we subtract three zero times. For the second guest, we subtract three once. For the third guest, we subtract three twice. So, for the nth guest, we're going to subtract three n minus one time. Notice, for the nth guest, we subtracted three twice. The second guest, subtracted three once. First guest, subtracted three zero times, so this works out. For the first guest, we would subtract three zero times, and so g of one would be 50. We can see that this is consistent for all of these. So I could write 50 minus three times n minus one, and I really recommend making the table here just so you make sure you get the n minus one or the n right and it all gels.