Arithmetic Series Can you add the first 100 counting numbers in less than 30 seconds? Begin How did he do it so quickly? It is said that he

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2 Little Freddie is said to have done the work in his head and written only the answer on his slate in less than 30 seconds. Can you do it in less than 30 seconds? Arithmetic Series An arithmetic series is the sum of the first n terms of an arithmetic sequence. As the story goes, little Freddie Gauss was 10 years old (some sources claim 7) in the late 1700s. There were no calculators or pencils. He worked with chalk on a slate. The teacher reportedly gave the class the task of adding the first 100 counting numbers. He expected the class to add Of course, he expected this to take a class of third-graders some time.

3 Arithmetic Series Can you add the first 100 counting numbers in less than 30 seconds? Begin How did he do it so quickly? It is said that he recognized this pattern (101) = 5050 Little Freddie Gauss was Carl Friedrich Gauss, one of history s most celebrated minds. He is credited with advancements in astronomy, physics, electromagnetism, math, and other fields. He is also largely responsible for developing the method of the sum of least squares.

4 Arithmetic Series Try the same technique on this problem: Find the sum of the first 60 positive even integers. 30(122) = 3660 S n means the sum of the first n terms of a series. Find S 10 for (-33) = -165 Write the formula for the sum of the first n terms in an arithmetic series.

5 Arithmetic Series Evaluate There is a difference between and = 3(2 + 12) = 42. = 41. Find S 10 for a 1 = 4, a 10 = 31, so S 10 = 175

6 Arithmetic Series Find S 34 for (-3) + To use we need the 34 th term. a 34 = 12 + (34 1)(-5) = -153 S 34 = 17( ) = Replace a n in with a n = a 1 + (n 1)d and simplify to get a new formula. Apply this new formula to S 34 above. Often, one can use a n = a 1 + (n 1)d to get the n th term and apply couple problems,. In some cases, like the next is needed.

7 Arithmetic Series A display in a supermarket is built with one can on top, two cans in the next row, and one more can in each succeeding row. If there are 171 cans in the display, how many cans are in the bottom row? 342 = n 2 + n 0 = n 2 + n n = 18 or cans in the bottom row

8 Arithmetic Series A portion of the wall of a log playhouse consists of 6 logs, each one 8 inches longer than the previous log. One 14-ft log is cut to create the 6 logs. Find the length of all 6 cut pieces. 168 = 3(2a 1 + (6-1)8) where a 1 is the shortest log and lengths are in inches. 168 = 3(2a ) 168 = 6a = 6a 1 a 1 = 8. Lengths are 8, 16, 24, 32, 40, and 48

9 Arithmetic Series In training for a marathon, an athlete plans to begin by running 2 miles each day the first week, and add a half-mile per day each week. How many miles will the runner have logged at the end of the 10 th week? S 10 = 5(4 + 9(0.5)) S 10 = 42.5 Recognize that this is the sum of the one-day distance from each of the 10 weeks This needs to be multiplied by 7 to include all days of the week. 42.5(7) = mi.

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11 Geometric Series Find the sum of This series is geometric. It is made up of the terms of a geometric sequence. Each term is a fourth of the previous term. There are only two missing terms, 50 and The sum is But what if 50 terms were missing instead of two terms? There must be an easier method, right? There is. It is somewhat like Gauss method with arithmetic series, but with geometric series, it s a little more complicated. It s demonstrated here with the example of grains of corn on the worksheet.

12 Geometric Series S 64 = S 64 = S 64 2S 64 = ( ) S 64 = S 64 = What does this equal? kernels Now try this formula from p.530:

13 Geometric Series Find the sum of using Find S 7 for

14 Geometric Series On the day she was born, and on each birthday thereafter, Jesse s parents deposited $200 in an account earning 7% annual interest. If there are no other adjustments except interest, how much is in the account after the deposit on her 18 th birthday? Birthday Value How many terms are there in 18 th 200 this geometric series? What 17 th 200(1.07) is r? What is g 1? 16 th 200(1.07) 2 1 st 200(1.07) 17 Birthday 200(1.07) 18

15 Geometric Series Years ago, my youngest son and I dropped a superball from a height of 5 in our kitchen and measured the bounce at 3 9. To what percent of the original height did the ball rebound? If with each bounce, it rebounds to 75% of the previous height, how far has the ball traveled when it strikes the ground the fourth time?

16 gives the sum of the 4 heights shown in the picture. 2( ) 5 gives the distance the ball travels Geometric Series If a ball is dropped from 5, and with each bounce it rebounds to 75% of the previous height, how far has the ball traveled when it strikes the ground the fourth time? Apply the formula for S n.

17 Review At noon, 1000 mg of medicine is administered to a patient. At the end of each hour, the concentration in the blood stream is 60% of the amount present at the beginning of the hour. What portion of medication remains in the blood stream at 4 pm? g 4 = 1000(.6) 4 = mg. If a second dose is administered at 2 pm, how much is in the blood stream immediately after the injection? 1360 mg

18 Review A student saving for college put away $100 on the first month, and increased that by a constant amount each month. On the 12 th month, she saved $925. What was the increase each month? 925 = d Recursive d = $75 Write a recursive and explicit formula for the situation. Explicit a n = (n 1) 75 Find the amount saved over the 12-month period. S 12 = 6( ) = $6150