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Understanding Geometric Sequences: Formulas, Examples, and Practices

This comprehensive guide explores geometric sequences, emphasizing their defining characteristics, such as the common ratio. Learn how to identify geometric sequences, determine common ratios, and apply the geometric sequence formula to find specific terms. With detailed examples, including finding the nth term of various sequences, and exercises on missing terms using geometric means, this resource equips learners with foundational knowledge and practice opportunities to grasp geometric sequences effectively. Perfect for students and educators alike!

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Understanding Geometric Sequences: Formulas, Examples, and Practices

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  1. Warm UpState the pattern for each step. • 3, 6, 12, 24, 48, 96,… • 81, 27, 9, 3, 1, ⅓,… • -2, 4, -8, 16, -32, 64, -128

  2. Geometric Sequences

  3. Geometric Sequences An geometric sequence is defined as a sequence in which there is a common ratio between consecutive terms. Common Ratio = 2

  4. Is the given sequence geometric? If so, identify the common ratio. • 5, 15, 45, 135, … • 15, 30, 45, 60, … • 6, -24, 96, -384, … • 8, 20, 32, 44, … • 1, 2, 4, 8, … • 7, 0.7, 0.07, 0.007, … • 10, 4, 1.6, 0.64, …

  5. Geometric Sequence Formula The 1st number in the sequence. The same as the n in an. If you’re looking for the 5th number in the sequence, n = 5. an = a1 • r (n-1) The “nth” number in the sequence. Ex. a5 is the 5th number in the sequence. The common ratio.

  6. Example 1: an = a1 • r (n-1) Given the sequence 4, 28, 196, 1372, 9604,…, find the 14th term.

  7. Example 2: an = a1 • r (n-1) Given the sequence -2, 6, -18, 54, -162,…, find the 17th term.

  8. Example 3: an = a1 • r (n-1) Given the sequence 100, 83, 68.89, 57.1787,…, find the 9th term.

  9. Example 4: an = a1 • r (n-1) Given the sequence 1, 5, 25, 125, 625, 3125,…, find the term number that is 9,765,625.

  10. Example 5: an = a1 • r (n-1) Suppose you want a reduced copy of a photograph. The actual length of the photograph is 10 in. The smallest size the copier can make is 64% of the original. Find the length of the photograph after five reductions.

  11. Geometric Mean • Used to find the missing term of a geometric sequence • The positive square root of the product of the two numbers

  12. Geometric Mean Ex 10: Find the missing term of each geometric sequence • 20, ____, 80, … • 3, ____, 18.75, … • 28, ____, 5103, … Solutions • 40 • 7.5 • 378

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