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Geometric Sequences

Geometric Sequences. A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.

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Geometric Sequences

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  1. Geometric Sequences A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number. • 1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it. • The multiplier from each term to the next is called the common ratio and is usually denoted by r.

  2. Finding the Common Ratio • In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it. The geometric sequence 1, 2, 4, 8, 16 , … has common ratio r = 2 since:

  3. Writing a Recursive Formula for a geometric sequence • The recursive formula is used to find the next term in the sequence • The recursive formula would just be as follows: the next term in the sequence (an) is equal to the previous term (an-1) times the common ratio (r) What would the recursive formulas for the following sequences be? (fill in the first term and the common ratio into the recursive formula format) a. 9, 81, 729 b. 81, 27,9 (hint: the common ratio will be a fraction)

  4. Writing an Explicit Formula for a geometric sequence • The explicit formula is used to find any term • in the sequence • The explicit formula would just be as follows: • the term that you are solving for (an) is • equal to the first term(a1) times the common ratio (r) • to the power of the previous term number • an= a1r(n-1) What would the explicit formulas for the following sequences be? (fill in the first term (a1) and the common ratio (r) into the recursive formula format) a. 4, -8, 16… (r is negative) b. 125, 25, 5… (r is a fraction)

  5. Using an Explicit Formula for Finding the nth Term Example Find a5 for the geometric sequence 4, –12, 36, –108 , … first term = 4 Common ratio = -3 Finding the Explicit Formula: an= a1r(n-1) an= (4)(-3)(n-1) Plugging in to find a5: a5= (4)(-3)(5-1) a5= 4 x (-3)4 a5 = 4 x 81 a5 = 324 You do: Find a6 for the following geometric sequence: 4, -8, 16…

  6. Geometric Sequence Word Problem ExampleA population of fruit flies grows in such a way that each generation is 1.5 times the previous generation. There were 100 insects in the first generation. How many are in the fourth generation. common ratio = 1.5 first term = 100 nth term = 4 Solution

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