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Worked example: arithmetic series (recursive formula)

Video transcript

so I'm gonna recursively define an arithmetic sequence so we're gonna say that the eighth term of the sequence is equal to the I minus one term of the secant sequence plus 11 plus 11 so the ithe so each term is going to be 11 more than the term before it now we have to establish a base case here and so we're gonna say that the first term of our arithmetic sequence is going to be equal to 4 so given this recursive definition of our arithmetic sequence right over here what I challenge you to do is to find the first 600 find the sum of the first 650 terms of the sequence then write that down to find the sum of first 650 terms of the sequence of this arithmetic sequence that we have just defined and like always pause the video and see if you can work that out alright so how can we think about this well certainly in many videos we gave our intuition for the the sum of an arithmetic sequence and we came up with a formula for evaluating in a sum of an arithmetic sequence which we'd call an arithmetic series and that sum of the first n terms is going to be the first term plus the last term over 2 so really the average of the first and the last term's times the number of terms we have and this is only the case when it's an arithmetic but it's an arithmetic series where each term that we're adding is a fixed amount larger or less than the term before it early we have a fixed difference so what about this one right over here what is what is the first and the last term going to be and what is our n well we know that n is 650 we know that n is 650 and we know what the first term is going to be the first term is going to be 4 but we need to figure out what the nth term is what is the or we need to figure out what the 650th term is going to be well let's think about this a little bit so this is going to be for taking the sum it's going to be four plus the next term this the the second term so if a sub two is going to be a sub one plus eleven so it's going to be four plus 11 which is 15 we're going to add 11 which is going to get us to 26 and we're going to keep a lot adding 11 now how many times are we going to add 11 well to get to the second term we add 11 once to get to the third term we add 11 twice to go so to get to the six hundred and fiftieth term so plus so this is a sub 650 a sub 650 we're going to have to add 11 look to get to the second term we added 11 once third term add 11 twice so to get to the six hundred and fiftieth term we are going to add 11 we're going to add 11 650 minus one times or six hundred and forty nine times notice to get to each term to get to the first term you added one minus one you had an 11 1 minus one times you added 11 0 times you start with the four I didn't add 11 at all then the second term you at 11 once third term you add 11 twice fort term you add 11 three times 650th term you at 11 649 times and so if you add 11 649 times what do you get so four plus six hundred 49 times 11 is going to be equal to I'll get my calculator out for this so this is going to be equal to six 49 times 11 is equal to now plus four is equal to seven thousand 143 seven thousand 143 so that's the six hundred and fiftieth term seven thousand one hundred and forty three and so now we can just evaluate this so we'll get the calculator out for that so we have seven thousand 143 plus four plus the first term plus four is equal to that we're gonna divide by two so divide by two it gets us three thousand five hundred seventy three point five we're gonna multiply that times six hundred fifty that's how many terms we have times six hundred fifty is equal to that's a pretty large number it's going to be equal to two million three hundred twenty two thousand seven hundred and seventy-five two million three hundred twenty two thousand seven hundred seventy five so I already forgot it three million I have to have a I have trouble remembering things when I take it off my screen all right two million three hundred two million three hundred twenty two thousand seven hundred and seventy-five I'm glad I had a calculator at hand for that one but you couldn't do it by hand I always encourage you to it never hurts to practice the arithmetic