It never hurts to practice the arithmetic. I'm glad I had a calculator in hand for that one, but you could do it by hand. In fact were going to do something extra special and use our friends from Python to guide us. I have trouble remembering things when I take it off my screen. Were going to talk about sequences, specifically about the relationship between closed forms and recursive formulas for sequences. Times 650 is equal to, that's a pretty large number, is going to be equal to 2,322,775. So we have 7,143 plus four plus the first term plus four is equal to that. And so if you add 11 649 times, what do you get? So four plus 649 times 11 is going to be equal to, I'll get my calculator out for this, so this is going to be equal to 649 times 11 is equal to, now plus four is equal to 7,143, 7,143. You started with the four, didn't add 11 at all. Notice, to get to each term, to get to the first term, you added one minus one, you had an 11 one minus one times, you added 11 zero times. So to get to the 650th term, we are going to add 11, we are going to add 11Ħ50 minus one times, or 649 times. So to get to the 650th term, 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. To get to the third term, we add 11 twice. Now how many times are we going to add 11? Well to get to the second term, we add 11 once. Which is going to get us to 26, and we're going to keep adding 11. If we're taking the sum, it's going to be four plus the next term, the second term, so if a sub two is going What the nth term is, or we need to figure out So what about this one right over here? 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. When it's an arithmetic series where each term that we're adding is a fixed amount larger or less than the term before it, or that the term in the sequence the common difference the term in the sequence the term number EX1: 7, 10, 13, 16. Create a recursive formula using the first term in the sequence and the common difference. The first and last terms, times the number of terms we have. Determine that the sequence is arithmetic. All right, so how can we think about this? Well, in many videos we give our intuition for the sum of an arithmetic sequence, and we came up with a formula for evaluating a sum ofĪn arithmetic sequence, which we call an arithmetic series, and that sum of the first n terms is going to be the first term plus the last term over two, so really the average of In this, the ratio between every two successive terms is equal to the same number. It is determined by the first term and the common difference. Arithmetic Sequence It is determined by the first term and the common ratio. And like always, pause the video and see if you can work that out. The recursive formula of a geometric sequence The sum of a finite geometric sequence. In most arithmetic sequences, a recursive formula is easier to create than an explicit formula. Find the sum of first 650 terms of the sequence, of this arithmetic sequence that we have just defined. Sequence right over here, what I challenge you to do is to find the sum of the first 650 terms of the sequence. So given this recursive definition of our arithmetic Say that the first term of our arithmetic sequence is going to be equal to four. Now we have to establish a base case here, and so we're going to So each term is going to be 11 more than the term before it. They ith term of the sequence is equal to the i minus oneth term of the sequence plus 11. \).To recursively define an arithmetic sequence.
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