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Arithmetic Sequences and Series

Arithmetic Sequences and Series. 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series. Arithmetic Sequences and Series. Objectives. Key Words. Arithmetic sequences The difference between consecutive terms is constant

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Arithmetic Sequences and Series

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  1. Arithmetic Sequences and Series 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

  2. Arithmetic Sequences and Series Objectives Key Words Arithmetic sequences The difference between consecutive terms is constant Common difference The constant difference of an arithmetic sequence is denoted by d. Arithmetic series The expression formed by adding the terms of an arithmetic sequence denoted by • Identify Arithmetic Sequences • Write rules and graph sequences • Find sums of series

  3. Example 1 Tell whether the sequence is arithmetic. a. 0, 7, 14, 21, 28 b. 1, 3, 6, 10, 15 SOLUTION To tell whether a sequence is arithmetic, find the differences of consecutive terms. a. a2 a177 0 b. a2 a132 1 – – – – = = = = a3 a2147 7 a3 a263 3 – – – – = = = = a4 a3217 14 a4 a3104 6 – – – – = = = = a5 a4287 21 a5 a4155 10 – – – – = = = = Identify Arithmetic Sequences

  4. Example 1 Identify Arithmetic Sequences Each difference is 7, so the sequence is arithmetic. The differences are not constant, so the sequence is not arithmetic.

  5. Checkpoint Tell whether the sequence is arithmetic. not arithmetic 1. 1, 8, 27, 64, 125 ANSWER arithmetic 2. 0, 2, 4, 6, 8 ANSWER arithmetic 3. 1, 4, 7, 10, 13 ANSWER Identify Arithmetic Sequences

  6. Identify Arithmetic Sequence • Rule for an Arithmetic Sequence • The nth term of an arithmetic sequence with first term and common difference d can be found using the following rule: And so on… Common difference The constant difference of an arithmetic sequence is denoted by d.

  7. SOLUTION The sequence is arithmetic with first term . The common difference is . So, a rule for the nth term is as follows. a1 20 = d 18 20 2 – – = = ( an a1 d – Write general rule. ( + n 1 = ( ( 20 Substitute for and for d. 20 a1 2 – – ( ( + n 1 –2 = 20 2n2 Use distributive property. – = + 22 2n Simplify. – = Example 2 Write a Rule for a Sequence Write a rule for the nth term of the arithmetic sequence 20, 18, 16, 14, . . . .Then find a9.

  8. ( ( 9 Example 2 Write a Rule for a Sequence You can find the 9th term by substituting 9 for n in the rule an 22 2n. The 9th term is a922 2 22 184. – – – = = = =

  9. One term of an arithmetic sequence is a1124. The common difference is d 2. = = a. Write a rule for the nth term. SOLUTION a. Find the first term. ( an a1 d – Write general rule. ( + n 1 = ( a11 a1 d – Substitute 11 for n. + 11 1 ( = ( 24 a1 Substitute 24 for a11 and 2 for d. + 10 2 ( = 24 a1 Multiply. + 20 = Example 3 Graph a Sequence

  10. 4 a1 Subtract 20 from each side. = ( an a1 d – ( Write general rule. + n 1 = ( 4 – ( Substitute. + n 1 2 = 42n2 – Distributive property = + 22n Simplify. = + Example 3 Graph a Sequence Write a rule for the nth term.

  11. Write Rules for Sequences and Graph Sequences Checkpoint 1; 25 an 2n ANSWER + = 4. Write a rule for the nth term of the sequence 3, 5, 7, 9, . . . . Then find a12.

  12. Find Sums of Series • The sum of a Finite Arithmetic Series • The sum of the first n terms of an arithmetic series is given by the following: • In words, the sum of the first n terms of an arithmetic series, , is the mean of the first and nth terms, multiplied by the number of terms. Arithmetic series The expression formed by adding the terms of an arithmetic sequence denoted by

  13. Find the number of cards in the first 20 rows of the house of cards on right. SOLUTION Notice that the first term is a13. The common difference is d 633. You need to write a rule for the nth term so that you can find the 20th term. = – ( = = an a1 d – Write general rule. ( + n 1 = ( 3 – ( + n 1 3 Substitute 3 for a1 and 3 for d. = 3n Simplify. = Example 4 Find the Sum of an Arithmetic Series

  14. Write rule for S20. Substitute 3 for a1 and 60 for a20. 360 + 20 = Simplify. 2 ANSWER 630 = There are 630 cards in the first 20 rows of the house of cards. a1a20 + S20 20 = 2 ( ( 20 Example 4 Find the Sum of an Arithmetic Series Find the 20th term. The 20th term is a20360. = = Find the sum of the first 20 terms.

  15. . . . . . . . . . . . . 155 ANSWER + + + + 7. 5 + 6 + 7 + 8 + 9 + 10 95 ANSWER 8. – 4 + 0 + 4 + 8 + 12 + 16 140 ANSWER ( 9. – 9 + + 5 + 12 + 19 ( ( 225 ANSWER – 2 ( Find Sums of Series Checkpoint Find the sum of the first 10 terms of the arithmetic series. 6. 2 + 5 + 8 + 11 + 14 + 17

  16. Conclusions Summary Assignment Arithmetic Sequences and Series Page 618 #(12,16,24,27-29, 34,37,40,48,49) • How can you tell that a sequence is arithmetic? • A sequence is an arithmetic sequence if the difference of consecutive terms is constant.

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