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Arithmetic Sequences and Series

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Sequences Series List with commas “Indicated sum” 3, 8, 13, 18 3 + 8 + 13 + 18

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An Arithmetic Sequence is defined as a sequence in which there is a common difference between consecutive terms.

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Which of the following sequences are arithmetic? Identify the common difference. YES YES YES NO NO

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T h e c o m m o n d i f f e r e n c e i s a l w a y s t h e d i f f e r e n c e b e t w e e n a n y t e r m a n d t h e t e r m t h a t p r o c e e d s t h a t t e r m. C o m m o n D i f f e r e n c e = 5

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The general form of an ARITHMETIC sequence. First Term: Second Term: Third Term: Fourth Term: Fifth Term: nth Term:

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Formula for the nth term of an ARITHMETIC sequence. I f w e k n o w a n y t h r e e o f t h e s e w e o u g h t t o b e a b l e t o f i n d t h e f o u r t h.

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Given: Find: IDENTIFYSOLVE

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Given: Find: What term number is -169? IDENTIFYSOLVE

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Given: IDENTIFYSOLVE Find: What’s the real question?The Difference

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Given: IDENTIFYSOLVE Find:

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A r i t h m e t i c S e r i e s

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W r i t e t h e f i r s t t h r e e t e r m s a n d t h e l a s t t w o t e r m s o f t h e f o l l o w i n g a r i t h m e t i c s e r i e s. W h a t i s t h e s u m o f t h i s s e r i e s ?

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71 + (-27) Each sum is the same. 50 Terms

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Find the sum of the terms of this arithmetic series.

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Find the sum of the terms of this arithmetic series. What term is -5?

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Alternate formula for the sum of an Arithmetic Series.

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Find the sum of this series It is not convenient to find the last term.

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An introduction………… Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms

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Find the next four terms of –9, -2, 5, … Arithmetic Sequence 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33

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Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k

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Vocabulary of Sequences (Universal)

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Given an arithmetic sequence with x 15 38 NA -3 X = 80

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-19 63 ?? x 6 353

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Try this one: 1.5 16 x NA 0.5

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9 x 633 NA 24 X = 27

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-6 29 20 NA x

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Find two arithmetic means between –4 and 5 -4, ____, ____, 5 -4 4 5 NA x The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence

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Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 5 4 NA x The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence

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Find n for the series in which 5 x y 440 3 X = 16 Graph on positive window

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Example: The nth Partial Sum The sum of the first n terms of an infinite sequence is called the nth partial sum.

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Example 6. Find the 150 th partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, …

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Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?

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Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation. So the total sales for the first 2o years is $1,625,000

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