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Using recursive formulas of geometric sequences

Video transcript

the geometric sequence a sub I is defined by the formula where the first term a sub one is equal to negative one-eighth and then every term after that is defined as being so a sub I is going to be two times the term before that so a sub I is two times a sub I minus one what is a sub four the fourth term in the sequence and pause the video and see if you can work this out well there's a couple of ways that you could tackle this one is to just directly use these formulas so we could say that a sub four well that's going to be this case right over here a sub four is going to be equal to two times a sub three well a sub three if we go and use this formula it's going to be equal to two times a sub two each term is equal to two times the term before it and then we can go back to this formula again and say a sub two is going to be two times a sub one two times a sub one and lucky for us we know that a sub 1 is negative one-eighth so it's going to be 2 times negative one-eighth which is equal to negative one-fourth negative 1/4 and so this is negative 1/4 so 2 times negative 1/4 is equal to negative 2/4 or negative 1/2 and so a sub 4 is 2 times a sub 3 a sub 3 is negative 1/2 so this is going to be 2 times negative 1/2 which is going to be equal to negative 1 so that's one way to solve it another way to think about it is look we have we have our initial term and we also know our common ratio we know each successive term is 2 times the term before it so we could we could explicitly this is a recursive definition for our geometric series but we could explicitly write it as a sub I is going to be equal to our initial term negative 1 8 and then we're going to multiply it by 2 we're going to multiply it by 2 I minus 1 times so we could say times 2 to the I minus one power let's make sure let's make sure if that let's make sure that makes sense so a sub one based on this formula a sub 1 would be negative 1/8 times 2 to the 1 minus 1/2 to the 0th power so that makes sense that would be negative 1/8 based on this formula a sub 2 would be negative 1/8 times 2 to the 2 minus 1 so 2 to the first power so we're gonna take our initial term and multiply it by 2 once which is exactly right a sub 2 is negative 1/4 and so if we want to find the fourth term in the sequence we could just say well using this explicit formula we could say a sub 4 is equal to negative 1/8 times 2 to the 4 minus 1/2 to the 4 minus 1 power and so this is equal to negative 1/8 times 2 to the 3rd power and so this is negative 1/8 times eight negative 1/8 one day give 1/8 times 8 which is equal to negative 1 and you might be a little bit a toss-up on which method you want to use but for sure this second method right over here where we come up with an explicit formula once we know the initial term and we know the common ratio this would be way easier if you were trying to find say the 40th term because doing the 40th sirum recursively like this would take a lot of time and frankly a lot of paper