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Algebra 2b Name: Arithmetic Series: The Sum of a Finite Arithmetic Sequence You have learned that an arithmetic sequence is a sequence with a common difference. For example: 5, 7, 9, 11, 13, 15, … If, as in this example, the sequence continues forever, we call it an infinite sequence. If the sequence stops, it is a finite sequence, such as 5, 7, 9, 11, 13, 15. Let’s now consider how to go about finding the sum of arithmetic sequences. 1. Explain why it is impossible to add up all the terms of an infinite arithmetic sequence. For example: 5 7 9 1113 15 ... It is possible to sum a finite arithmetic sequence. The sum of any number of terms of a sequence is called a series. For example, 5 7 9 1113 15 is a series. The value of this series is 60. 2. Find the value of the series 1 2 3 4 5 6 7 8 9 10. While we can always just add up the terms, evaluating a series that has a large number of terms by adding themall up is annoying and time consuming. For example, consider 1 2 3 4 5 ... 95 96 97 98 99 100 . For this reason, it is worth the effort to find an easier way. Let’s use 1 2 3 4 5 6 7 8 9 10 to investigate this idea. 3. Find the sum of the first term and the last term: 110 4. Find the sum of the second term and the second to last term: 2 9 5. Find the sum of the third term and the third to last term: 3 8 6. What do you notice? Will this be true for the remaining pairs? 7. How many such pairs are there? 8. There are _____ pairs and each pair adds up to _____. Therefore the sum of the series is _____ times _____ : ___ ___ ___ Does this match the sum you found in question 2? 9. Evaluate the series 7 14 21 28 35 42using addition. 10. What is the sum of the first and last term? The second and second to last term? How many suchpairs are there? What then is the sum? Does this match what you found in question 9? 11. Make a conjecture for a formula for the sum of a finite arithmetic series. 12. Try your formula on the following series: 3 13 23 33 43 53 13. Does your formula work if there is an odd number of terms? Try this: 9 16 23 30 37 Hint: How many pairs are there? 14. How about if the terms are decreasing? Try this: 14 9 4 1 6 11 16 15. Does your formula work if the series is not arithmetic? Try this: 2 4 8 16 32 64 128 16. Find the sum of the first 35 terms of the sequence an 6n 2 . 17. Find the sum of the series 10 19 28 37 ... 478 487 496 505 18. Once you’re sure your formula works, try to write your formula using the correct mathematical symbols: The first term is a1, and the last term is an , where n is the number of terms. The sum of a finite arithmetic series =