Using arithmetic sequences formulas

all right we're told that the arithmetic sequence a sub I is defined by the formula where the eighth term in the sequence is going to be four plus three times I minus one what is a sub 20 and so a sub 20 is the 20th term in the sequence and I encourage you to pause the video and figure out what is the 20th term well we can just think about it like this a sub 20 we just use this definition of the I term everywhere we see an i we would put a 20 in so it's going to be 4 plus 3 times 20 minus 1 so once again just to be clear a sub 20 where instead of a sub i wherever we saw an i we put 20 and now we can just compute what this is going to be equal to this is going to be equal to 4 plus 3 times 20 minus 1 is 19 3 times 19 let's see 3 times 19 is 57 right is 30 plus 27 yep this is 57 and 4 plus 57 is equal to 61 so the 20th term in this arithmetic sequence is going to be 61 let's do another one of these and here they've told us the arithmetic sequence a sub I is defined by the formula a sub 1 to give us the first term and they say every other term so a sub I they're defining it in terms of the previous term so a sub I is going to be a sub I minus 1 minus 2 so this is actually a recursive definition of our arithmetic sequence so let's see what we can make of this so a sub 5 a sub 5 is going to be equal to we'll use this second line right here a sub 5 is going to be equal to a sub 4 minus 2 but we don't know what a sub 4 is just yet so let's try to figure that out so we could say that a sub 4 is equal to well if we use the second line again it's going to be a sub 3 minus 2 we still don't know what a sub 3 is I'll keep switching colors because it looks nice a sub 3 is going to be equal to a sub 3-1 so a sub 2 minus 2 we don't we still don't know what a sub 2 is and so we could write a sub 2 is equal to a sub 2 minus 1 so that's a sub 1 minus 2 now luckily we know what a sub 1 is a sub 1 is negative 7 a sub 1 is negative 7 so if this is negative 7 then a sub 2 is negative 7 minus 2 which is equal to negative 9 well that starts helping us out because if a sub 2 is negative 9 if this is negative 9 then a sub 3 is negative 9 minus 2 which is equal to negative 11 well now that we know that a sub 3 is negative 11 so this is negative 11 we could figure out a sub 4 is negative 11 minus 2 which is equal to negative 13 and we're almost there we know a sub 4 the fourth term in this arithmetic sequence is negative 13 so we can now so this is negative 13 a sub 5 is going to be a sub 4 which is negative 13 minus 2 which is equal to negative 15 so the fifth term in the sequence is negative 15 and we're all done