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1. Grade 10 – Mathematics
Quarter I
INFINITE GEOMETRIC SERIES

2. The sum of an infinite geometric series is given by:
𝑺∞ =
𝒂 𝟏
𝟏 − 𝒓
𝐰𝐡𝐞𝐫𝐞 − 𝟏 < 𝐫 < 𝟏

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. 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. 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. 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. 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. 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. 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. 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. 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. 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.