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# Infinite Geometric Series

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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

For more instructional resources CLICK me here and please DON'T FORGET TO SUBSCRIBE. 
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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.