This document provides instruction on finding the sum of finite geometric sequences. It defines a geometric series as the sum of terms in a geometric sequence. It gives examples of finding the sum of the first n terms when the ratio r is -1, 1, or another value. The key formula provided is Sn = a1rn-1/(r-1) for finding the sum of a finite geometric sequence, where a1 is the first term, r is the common ratio, and n is the number of terms. An example problem applies this to find the total distance traveled by a ball bouncing repeatedly to 40% of its previous height.

1. Grade 10 – Mathematics
Quarter I
GEOMETRIC SERIES AND FINDING THE
SUM OF FINITE GEOMETRIC SEQUENCE

2. Objectives:
•define geometric series;
•find the sum of the finite geometric sequence;
and
•solve problem involving geometric series.

3. A geometric series is the
sum of the terms of a
geometric sequence.

4. What is the sum of the first 10 terms of 2+2+2+…?
Solution 2+2+2+2+2+2+2+2+2+2 =10(2) =20
What if 𝑟 = −1?
If 𝑟 = −1 and 𝑛 is even, then 𝑆 𝑛 = 0.
If 𝑟 = −1 and 𝑛 is odd, then 𝑆 𝑛 = 𝑎1.
If 𝑟 = 1, then 𝑆 𝑛 = 𝑛𝑎1.

5. Example: What is the sum of first 10
terms of 2 – 2 +2 – 2 + …?
𝑟 = −1
𝑛 is even
𝑆10 = 𝟎

6. Example: What is the sum of first 11
terms of 2 – 2 +2 – 2 + …?
𝑟 = −1
𝑛 is odd
𝑆11 = 𝟐

7. To find the sum of finite geometric
sequence,
𝑆 𝑛 =
𝑎1 𝑟 𝑛
− 1
𝑟 − 1
𝑟 ≠ 1

8. What is the sum of the first five terms
of 3, 6, 12, 24, 48, 96, …?
Solution 𝑎1 = 3, 𝑟 = 2, 𝑛 = 5
𝑆 𝑛 =
𝑎1 𝑟 𝑛 − 1
𝑟 − 1
=
3 (2)5
−1
2 − 1
=
3 32 − 1
1
𝑆5 = 𝟗𝟑
𝑟 =
6
3
= 2

9. A ball is tossed to a height of 4 meters rebounds to 40%
of its previous height. Find the distance the ball travelled
when it strikes the ground for the fifth time.
↑ 4𝑚 +↓ 4𝑚 = 8 𝑚
↑ 1.6𝑚+ ↓ 1.6𝑚 = 3.2𝑚
4 0.4 = 1.6
1.6 0.4 = 0.64
↑ 0.64𝑚+ ↓ 0.64𝑚 = 1.28𝑚
8 + 3.2 + 1.28 + ⋯
𝑎1 = 8, 𝑟 =
3.2
8
= 0.4, 𝑛 = 5
𝑆 𝑛 =
𝑎1 𝑟 𝑛
− 1
𝑟 − 1
=
8 (0.4)5
−1
0.4 − 1
=
−7.91808
−0.6
𝑆5 = 𝟏𝟑. 𝟏𝟗𝟔𝟖
𝑆5 = 𝟏𝟑. 𝟐𝟎 𝒎