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# Recursive formulas for arithmetic sequences

CCSS.Math:

## Video transcript

G is a function that describes an arithmetic sequence here are the first few terms of the sequence so they say the first term is 4 second term is 3 and four-fifths third term is 3 and three-fifths fourth term is 3 and 2/5 find the values of the missing parameters a and B and the following recursive definition of the sequence so they say the nth term is going to be equal to a if n is equal to 1 and it's going to be equal to G of n minus 1 plus B if n is greater than 1 and so I encourage you to pause this video and see if you can figure out what a and B are going to be well the first one to figure out a is actually pretty straightforward if n is equal to 1 if n is equal to 1 the first term when N equals 1 is 4 so a is equal to 4 so we could write this as G of n is equal to 4 if n is equal to 1 and now let's think about the second line the second line is interesting it's saying it's going to be equal to the previous term G of n minus 1 this is this means the n minus 1 term plus B will give you the nth term let's just think about what's happening with this arithmetic sequence when I go from the first term to the second term what have I done I have looks like I've subtracted 1/5 so minus 1/5 and then it's an arithmetic sequence so I should subtract or add the same amount every time and I am I'm subtracting 1/5 and so I am subtracting 1/5 and so one way to think about it if we were to go the other way we could say for example that G of 4 is equal to G of 3 minus 1/5 minus 1/5 you see that right over here G of 3 is this you subtract 1/5 you get G of 4 you see that right over there and of course I could have written this like G of 4 is equal to G of 4 minus one minus one-fifth so when you look at it this way you can see that if I'm trying to find the nth term it's going to be the N minus one term plus negative one-fifth so B is negative one-fifth once again if I'm trying to find the fourth term if n is equal to four no I'm not going to use this first case because this has to be for N equals one so if N equals four I would use the second case so then it would be G of four minus one it would be G of 3 minus one-fifth and so we could say G of n is equal to G of n minus one so the term right before that minus one-fifth if n is greater than one but for the sake of this problem we see we see that a is equal to four and B is equal to negative one-fifth