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Geometric Series and Finding the Sum of Finite Geometric Sequence

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Geometric Series and Finding the Sum of Finite Geometric Sequence

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Geometric Series and Finding the Sum of Finite Geometric Sequence

1. 1. Grade 10 – Mathematics Quarter I GEOMETRIC SERIES AND FINDING THE SUM OF FINITE GEOMETRIC SEQUENCE
2. 2. Objectives: •define geometric series; •find the sum of the finite geometric sequence; and •solve problem involving geometric series.
3. 3. A geometric series is the sum of the terms of a geometric sequence.
4. 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. 5. Example: What is the sum of first 10 terms of 2 – 2 +2 – 2 + …? 𝑟 = −1 𝑛 is even 𝑆10 = 𝟎
6. 6. Example: What is the sum of first 11 terms of 2 – 2 +2 – 2 + …? 𝑟 = −1 𝑛 is odd 𝑆11 = 𝟐
7. 7. To find the sum of finite geometric sequence, 𝑆 𝑛 = 𝑎1 𝑟 𝑛 − 1 𝑟 − 1 𝑟 ≠ 1
8. 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. 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