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CCSS.Math: , , , , , ,

mohamad 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 Muhammad's backyard in the NT 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 one point four might say well each successive term we're multiplying or dividing by the same number well it's going to be geometric let's make that a little bit more tangible just just in case so let's make a little tent 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 1 the first year n equals 1 there were 500 leaves F of n is 500 now what n is equal to 2 we're going to grow by 40% which is the same thing as multiplying by 1.4 so 500 times 1.4 let's see 40 percent of 500 is 200 so we're going to grow by 200 so we're going to go to 700 then in year 3 we're going to grow by 40 percent of 700 which is 280 so it's going to grow to 980 so notice it's definitely not an arithmetic it's definitely not a arithmetic sequence an arithmetic sequence we would be adding or subtracting the same amount every time but we're not here from 507 or we grew by 200 and then from 709 80 we grew by 280 instead we're multiplying or dividing by the same amount each time in this case we're multiplying by one point four by one point for 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 asked 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 1 well we know that when N equals 1 we had 500 leaves on the trees Oh 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're going to have let's see it's going to be the previous year the previous year times what well 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 B is going to be 1.4 you take the previous year and you multiply by 1.4 for any other year a year other than n equals 1 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 C uun who 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 Cu Ewan had before the nth guests arrived all right actually before I even look at these questions let me make a table here because they're saying before the antha cast I want to make sure I'm understanding this properly so so this is N and then this is going to be G of n right over here so when n is equal to 1 when n is equal to 1 G of n is going to be or G of 1 is going to be the number of party favors cuu and had before the first guests well before the first guests 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 3 to the first guests so she's now going to have 47 party favors now when n is equal to 3 how many party favors did she have before the third guest well she would have had to give party favors to the first and second guess who each got 3 so she would have 44 and I think you see the pattern for every time and when N equals 1 G of n is 50 and every time we increase n by 1 every time we we increment n we are increasing G of n by plus 3 by minus three I should say 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 and 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 guess we subtract three once for the third guest we subtract three twice so for the nth guess we're going to subtract three n minus one times notice for the amp guess we subtracted three twice the second guess subtracted three once first guess subtracted three zero times so this works out for the first guess 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 in all gels