Students should realize that no, this is not an actual running time but acts as a placeholder so that on June 1st the equation amounts to 15 minutes. Deriving the formula to calculate the sum of n terms, given an arithmetic series:. When using the a(0) term, ask students if there was ever a day Mallory ran 10 minutes. I ask students when we want to start adding the five minutes, and they are able to reason that this is not until June 2nd, thus creating the need to “back track” our equation. Make sure students understand the necessity of the (n-1) when starting with June 1st. In the debrief, highlight the fact that there are two ways to write the explicit formula (and in fact, there are infinitely many, since we could use any day in June as our “anchor”). Be ready to build on student thinking and use the debrief to discuss both methods. Both of these strategies lead to the same sum formula, though written slightly differently. This sum of 175 will occur 15 times since there are 15 pairings of days. Another strategy is to realize that the days can be summed in any order and the sum of the first and last day is the same as the sum of the second and second to last day, is the same as the sum of the third and third to last day, and so on. Students use the idea of her average run time to find the sum of all 30 days. This idea of a constant (common) difference is critical to the rest of this lesson and ties in important ideas about a constant rate of change and linear functions. We specifically ask for June 29th so students recognize that her running time on that day is exactly five less than her running time on the 30th. While students may use a recursive pattern to find the first few values in the table, they should quickly recognize the need to make use of structure to find values for days later in June. Students identify that her time increases by five minutes every day and use this to fill in her running log. You can also calculate the sequence of n th partial sums, which appears to diverge also, meaning the series diverges.Today students look at Mallory’s running times during the month of June to explore the idea of arithmetic sequences. Solution: Look at the terms in the series:īecause the terms are increasing in size as n approaches ∞, the series does not converge (i.e., it diverges). Practice Problem: Determine if the series converges. to find the next term in the series is the sum of a given number of terms in the. relationship that defines some underlying property of the series and can be used. Since a series is a sum of a sequence, the terms in a series also have a special. If it is, find the common ratio, the 8th term, and the explicit formula. a series is the sum of all terms from a1 to an. K Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name Introduction to Sequences Date Period Find the next three terms in each sequence. Worksheet by Kuta Software LLC Kuta Software - Infinite Precalculus Geometric Sequences and Series Name Date Period-1-Determine if the sequence is geometric. It is important to simply note that divergence or convergence is an important property of both sequences and series-one that will come into play heavily in calculus (particularly integral calculus). Y 6 wAWlslA wruiPg xhtOs9 3rSeIsoe4rIv Ye0d L.I i 9MOavd Jex AwdiztFhP uIGnvf Si0ngi ot Wes KAYlGgre Kbkr 6av B2U. Here again, we will not get into the mathematical machinery for proving convergence or divergence of a series. Interestingly, then, note that some series-even though they have an infinite number of terms-still converge. To close, let's consider a couple other series. Since this sequence obviously diverges, so does the series. This is clear in the above case: this sequence is Coincidentally in the case of the natural numbers, the domain and range are identical (assuming the first index value is 1-an assumption that we will stick with here).Īs a more concise representation, we can express the general sequence above as of nth partial sums for a series diverges, then so does the series. The range of this function is the values of all terms in the sequence. Although this construct doesn't look much like a function, we can nevertheless define it as such: a sequence is a function with a domain consisting of the positive integers (or the positive integers plus 0, if 0 is used as the first index value). The variables a i (where i is the index) are called terms of the sequence. More broadly, we can identify an arbitrary sequence using indexed variables: This ordered group of numbers is an example of a sequence. Relate convergence of a sequence to the concept of a limitĬonsider the natural numbers, a portion of which are shown below.
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