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Mastering Geometric Sequences and Series

Learn about geometric sequences, find terms and sums, master common ratios, and solve examples in this comprehensive guide.

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Mastering Geometric Sequences and Series

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  1. Chapter 12 Sequences and Series

  2. 12.3 Geometric Sequences and Series

  3. Geometric Sequences and Series • a1 • a2 • a3 • a4 • an • a1 • a1*r • a1*r*r • a1 *r*r*r • a1 • a1*r • a1*r2 • a1 *r3 • a1*rn-1 • Geometric Sequences have “r” the common ratio! • If the first term is 3 and the common ratio is 2 find the first four terms • 3, 6, 12, 24

  4. Finding the nth term of a Geometric Sequence: • an = a1rn-1 • a1 = first term • r = common ratio: a2 / a1 • n = term we are looking for

  5. Examples • Find the tenth term of 2, -10, 50 … • -3,906,250 • Find the tenth term of 1, 1/3, 1/9 … • 1/19,683 • Find the eight term of 5, 15, 45 … • 10,935

  6. Examples Cont. • If the third term of a geometric sequence is 63/4 and the sixth term is 1701/32 find the fifth term. • Hint a6 =a3*r3 • Therefore: 1701/32=63/4 r3 • And r = 3/2 • And the fifth term is 567/16

  7. Sum of a Geometric Series (Finite) • Find the sum of the first 10 terms: 1+0.7+0.49… • Approximately 3.2392… • Find the sum of the first 20 terms: 2 + -4 + 8 + -16 … • -699,050

  8. Examples Cont: • 7 + -14/3 + 28/9 … • 931/243

  9. Infinite Sum • Find the sum: 2 + 2/5 + 2/25 + … • 5/2 • Proof: write 0.33333… as a fraction • Write .454545… as a fraction • 5/11 • Write 2.351515151… as a fraction • 388/165

  10. What about?

  11. Homework • Pg 839-841: 1 – 5, 7 – 12, 24, 26, 30, 38 – 42 even, 45 - 47

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