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Identifying Patterns in Sequences and Series: Next Numbers and Summation Notation

This lesson focuses on recognizing patterns in sequences and series, helping students determine the next three numbers in given sequences. It covers the definition of sequences as functions of consecutive integers and introduces concepts like terms of a sequence, summation notation, and the process of finding sums in series. Students will engage in warm-up questions and practice identifying rules governing sequences while honing their calculation skills for sums of series.

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Identifying Patterns in Sequences and Series: Next Numbers and Summation Notation

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  1. Warm-Up Question Find next three numbers of each list of numbers by identifying the pattern. 1, 2, 3, 4, … 2, 4, 6, 8, … 5, 8, 11, 14, … 14, 10, 6, 2, … 2, 4, 8, 16, … 27, 9, 3, 1, … 1, 1, 2, 3, 5, 8, … 1, 3, 7, 15, 31, …

  2. Sequences and Series Definition of Sequences and Series

  3. Definition of Sequences Lesson: P.434 Sequence is a function whose domain is a set of consecutive integers starting with 1 unless specified. The values in the range are called the terms of the sequence. Domain: 1 2 3 4 5 6 … n Position of each term Range: Terms of the sequence Examples) 1 2 3 4 5 6 … 100 1 2 3 4 5 6 … 100 1 2 3 4 5 6 … 100 2 4 6 8 10 12 … 200

  4. Terms of sequences Lesson: P.434 Write the first four terms of each sequence. Required Practice: P.434 G.P. 1, 2, 3 Additional Practice: P.438 7, 9, 11, 13

  5. Rules of Sequences Lesson: P.435 Describe the pattern, write the next term, and write a rule for the nth term of each sequence. a) -1, -8, -27, -64 b) 0, 2, 6, 12 Required Practice: P.438 15, 16 Additional Practice: P.438 19, 20

  6. Warm-Up Question (9C) Describe the pattern, write the next term, and write a rule for the nth term of each sequence. 1, 3, 5, 7, 9, … 4, 7, 10, 13, 16, … 5, 1, -3, -7, -11, … 13, 11, 9, 7, 5, …

  7. Warm-Up Question (9AB) Calculate each sum. 1+ 2 = ? 1 + 2 + 3 = ? 1 + 2 + 3 + 4 = ? 1 + 2 + 3 + 4 + 5 = ? 1 + 2 + 3 + 4 + 5 + … + 99 + 100 = ? 2 + 4 = ? 2 + 4 + 6 = ? 2 + 4 + 6 + 8 = ? 2 + 4 + 6 + 8 + 10 = ? 2 + 4 + 6 + 8 + 10 + … + 98 + 100 = ?

  8. Definition of Series Lesson: P.434 When the terms of a sequence are added together, the resulting expression is a series. Sequence: 1 2 3 4 5 6 … n 2 4 6 8 10 12 … n 1 + 2 + 3 + 4 + 5 + 6 + … + n 2 + 4 + 6 + 8 + 10 + 12 + … + n Series:1 3 6 10 15 21 ... 2 6 12 20 30 32 …

  9. Summation Notation Lesson: P.436 Summation notation is often used to write a series. 4: Upper limit of summation 2i: Rule for each term. ∑: Greek letter Sigma to notate summation = 2 + 4 + 6 + 8 1: Lower limit of summation i: Index of summation Ex) Write the series using summation notation. a. 25 + 50 + 75 + … + 250 b. Required Practice: P.436 6, 7, 8, 9 Additional Practice: P.438 39, 40, 43

  10. Sum of a Series Lesson: P.437 Find the sum of the series Required Practice: P.437G.P. 10, 11 Additional Practice: P.439 45, 47, 49

  11. Formula for Special Series Lesson: P.437 Sum of n terms of 1 Sum of first n positive integers Sum of squares of first n positive integers Ex) Find the sum of the series Required Practice: P.437G.P. 12, 13 Additional Practice: P. 439 53, 55, 56

  12. Homework Assignment

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