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The concept of geometric sequences, including the definition, common ratio, and formula for finding the nth term. It provides examples for determining if a sequence is geometric and generating the first five terms of a sequence. Additionally, it explains how common exponents can be used to solve equations involving geometric sequences.
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Math 2200
Geometric Sequence: a sequence in which the ratio of consecutive terms is constant.
Common ratio: the ratio of successive terms in a geometric sequence. The ratio may be positive or negative. 𝑟 =
For example, for the following sequence, 2, 4, 8, 16, … , the ratio is:
Example 1: Determine if each sequence is geometric:
(A) 3, 9, 27, …
To develop the formula for a geometric sequence, consider a sequence such as 1, 3, 9, 27, 81, ….
i. What is the common ratio?
ii. Can this sequence be rewritten to show the pattern of the first term and the common ratio
iii. Can you write the pattern in general terms for any first term and common ratio?
Example 4: The first three terms of a geometric sequence are {𝑥 − 1 , 2𝑥 , 3𝑥 + 9 , … }^. Algebraically determine the value of 𝑥.
Common Exponents If an exponential equation has the same exponent, the bases must be equal as well. We can use this fact to help us solve equations involving geometric sequences. We can also use the properties of 𝑛th^ roots.
Example 5: Determine the formula for the 𝑛th^ term: 𝑡 3 = 5, 𝑡 6 = 135
Example 7: In a geometric sequence, the second term is 28 and the fifth term is 1792. Determine the values of 𝑡 1 and 𝑟, and list the first three terms of the sequence.
Textbook Questions: page: 39 - 44; # 1, 3, 5, 6, 7, 10, 14, 16, 23, 25
