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Infinite Geometric Series
The document covers the concept of infinite geometric series, including the formula for their sum and several examples illustrating how to calculate these sums. It discusses the conditions under which the sums exist based on the common ratio, r, and explains cases where the sum diverges. Additionally, it references educational materials related to grade 10 mathematics.
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Infinite Geometric Series
- 1. Grade 10 –Mathematics Quarter I INFINITE GEOMETRIC SERIES
- 2. The sum ofan infinite geometric series is given by: 𝑺∞ = 𝒂 𝟏 𝟏 − 𝒓 𝐰𝐡𝐞𝐫𝐞 − 𝟏 < 𝐫 < 𝟏
- 3. Find the sumof 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. Find the sumof 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. Find the sumof 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. Find the sumof 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. Tell whether ifthe sum exist in the following infinite geometric series 𝟑 + 𝟗 + 𝟐𝟕 + 𝟖𝟏 + ⋯ 𝑟 = 9 3 = 3 No, since r > 1.
- 8. Tell whether ifthe 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. Show that therepeating 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. Show that therepeating 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. The sum toinfinity 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. 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.