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Infinite Geometric Series

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References:
Nivera, G. C. (2015), Grade 10 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.

Mathematics Grade 10 Learner's Module (2015). Department of Education

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References:
Nivera, G. C. (2015), Grade 10 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.

Mathematics Grade 10 Learner's Module (2015). Department of Education

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Infinite Geometric Series

  1. 1. Grade 10 – Mathematics Quarter I INFINITE GEOMETRIC SERIES
  2. 2. The sum of an infinite geometric series is given by: 𝑺∞ = 𝒂 𝟏 𝟏 − 𝒓 𝐰𝐡𝐞𝐫𝐞 − 𝟏 < 𝐫 < 𝟏
  3. 3. Find the sum of each infinite geometric series. 𝟔𝟒, 𝟏𝟔, 𝟒, 𝟏, … 𝑟 = 16 64 = 1 4 𝑎1 = 64 𝑺∞ = 𝒂 𝟏 𝟏 − 𝒓 Given: = 𝟔𝟒 𝟏 − 𝟏 𝟒 = 𝟔𝟒 𝟑 𝟒 = 𝟔𝟒 ∙ 𝟒 𝟑 = 𝟐𝟓𝟔 𝟑
  4. 4. Find the sum of each infinite geometric series. 𝟏 𝟑 + 𝟏 𝟗 + 𝟏 𝟐𝟕 + 𝟏 𝟖𝟏 + ⋯ 𝑟 = 1 9 1 3 = 1 9 ∙ 3 1 = 3 9 = 1 3 𝑎1 = 1 3 𝑺∞ = 𝒂 𝟏 𝟏 − 𝒓 Given: = 1 3 𝟏 − 𝟏 𝟑 = 1 3 𝟐 𝟑 = 1 3 ∙ 𝟑 𝟐 = 𝟑 𝟔 = 𝟏 𝟐
  5. 5. Find the sum of each infinite geometric series. −𝟒, −𝟏, − 𝟏 𝟒 , − 𝟏 𝟏𝟔 … 𝑟 = −1 −4 = 1 4 𝑎1 = −4 𝑺∞ = 𝒂 𝟏 𝟏 − 𝒓 Given: = −4 𝟏 − 𝟏 𝟒 = −4 𝟑 𝟒 = −4 ∙ 𝟒 𝟑 = − 𝟏𝟔 𝟑
  6. 6. Find the sum of each infinite geometric series. 1 + 𝟐 + 𝟐 + 𝟐 𝟐 + ⋯ 𝑟 = 2 1 = 2 The sum does not exist since r > 1. Given: 2 = 1.41
  7. 7. Tell whether if the sum exist in the following infinite geometric series 𝟑 + 𝟗 + 𝟐𝟕 + 𝟖𝟏 + ⋯ 𝑟 = 9 3 = 3 No, since r > 1.
  8. 8. Tell whether if the sum exist in the following infinite geometric series −𝟐 𝟑 + 𝟐 𝟗 − 𝟐 𝟐𝟕 + 𝟐 𝟖𝟏 − ⋯ 𝑟 = 2 9 −2 3 = 2 9 ∙ − 3 2 Yes, since r < 1. 1 1 1 3 = − 1 3
  9. 9. Show that the repeating decimals 𝟎. ഥ𝟔 equals 𝟐 𝟑 𝟎. ഥ𝟔 = 𝟎. 𝟔𝟔𝟔𝟔 … 𝑟 = 6 100 6 10 = 6 100 ∙ 10 6 = 60 600 = 1 10 Given: 𝑎1 = 6 10 = 𝟔 𝟏𝟎 + 𝟔 𝟏𝟎𝟎 + 𝟔 𝟏𝟎𝟎𝟎 + 𝟔 𝟏𝟎𝟎𝟎𝟎 + ⋯
  10. 10. Show that the repeating decimals 𝟎. ഥ𝟔 equals 𝟐 𝟑 𝟎. ഥ𝟔 = 𝟎. 𝟔𝟔𝟔𝟔 … 𝑟 = 1 10 Given: 𝑎1 = 6 10 𝑺∞ = 𝒂 𝟏 𝟏 − 𝒓 = 6 10 𝟏 − 𝟏 𝟏𝟎 = 6 10 𝟗 𝟏𝟎 = 6 10 ∙ 𝟏𝟎 𝟗 = 𝟔 𝟗 = 𝟐 𝟑
  11. 11. The sum to infinity of a geometric series is twice the first term. What is the common ratio? 𝒂 𝟏 𝟏 − 𝒓 = 𝟐𝒂 𝟏 𝒂 𝟏 = 𝟏 𝟏 𝟏 − 𝒓 = 𝟐(𝟏) 𝟏 𝟏 − 𝒓 = 𝟐 𝟏 = 𝟐 − 𝟐𝒓 𝟏 = 𝟐(𝟏 − 𝒓) 𝟏 − 𝟐 = −𝟐𝒓 -1= −𝟐𝒓 r = 𝟏 𝟐
  12. 12. REFERENCES: ❖ Nivera, G.C. (2015). Grade 10 Mathematics Pattern and Practicalities. Don Bosco Press, Inc. Makati City, Philippines. ❖ Mathematics Grade 10 Learner’s Module. Department of Education. Pasig City, Philippines.

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