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Geometric Sequences

Dive into the world of geometric sequences with this comprehensive guide. Learn how to identify the general term of a sequence using the formula (a_n = a_1 cdot r^{n-1}), where (r) is the common ratio. Explore examples, including calculating specific terms like the 12th and 25th terms of given sequences. We also cover summation formulas to find totals across terms. Apply your knowledge through real-world scenarios, such as salary increases based on geometric progression. Perfect for students and anyone interested in mathematics!

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Geometric Sequences

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  1. Geometric Sequences Part 2

  2. Geometric Sequences • Geometric Sequences are found by multiplying the previous term by the same number, r. • The general term is found using: • an = a1 • rn-1

  3. Geometric Sequences • Find the 12th term of the sequence -1, 4, -16, 64, … • Find the general term first using the formula, then use the general term to find the specified term. • an = a1 • rn-1 • r = -4 and a1 = -1 • an = -1 • (-4)n-1 • a12 = -1 • (-4)11 = 4,194,304

  4. Geometric Sequences • You try a couple now: • Find the 25th term of the sequence -7, -14, -28, -56, … • -117,440,512 • Find the 7th term of the sequence 9, 3, 1, , …

  5. Geometric Sequences • Summation • Now we will use the summation formula for geometric sequences. • Sn = a1() • r = is the common ratio • n = the number of terms • a1 = the first term in the sequence

  6. Geometric Sequences • Let’s start with the easy ones: • 4 + 8 + 16 + 32 + … + 4(212) • a1 = 4 • r = 2 • n = 13 • Sn = a1() • = 4() • = 4() • = 32,764

  7. Geometric Sequences • Now to the ones that you are more familiar with: • k – 1 • a1 = 3 • n = 13 • r = 0.4 • Sn = a1() • = 3() • = 3() • ≈ 4.99997

  8. Geometric Sequences • Your turn!! • 1 + 2 + 4 + 8 + … + 1(28) 511 • 1.9998

  9. Geometric Sequences • Application time…..yahoo! • Sarah is offered a government job with the Department of Transportation. She is hired on the GS scale at a base rate of $32,000 with 1.7% increases in her salary per year. Calculate what her salary will be after she has been with the department for 12 years. • Find the general term….. • an = a1• rn-1 • a1 = 32000 • r = 1.017 • an = 32000 • 1.017n-1 • n = 12 • = 32000 • 1.01711 • = $38,519.48

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