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11.3 – Geometric Sequences

11.3 – Geometric Sequences. What is a Geometric Sequence?. In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio . Unlike in an arithmetic sequence, the difference between consecutive terms varies.

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11.3 – Geometric Sequences

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  1. 11.3 – Geometric Sequences

  2. What is a Geometric Sequence? • In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio. • Unlike in an arithmetic sequence, the difference between consecutive terms varies. • We look for multiplication to identify geometric sequences.

  3. Ex: Determine if the sequence is geometric. If so, identify the common ratio • 1, -6, 36, -216 yes. Common ratio=-6 • 2, 4, 6, 8 no. No common ratio

  4. Important Formulas for Geometric Sequence: • Recursive Formula • Explicit Formula an = (an– 1 ) r an = a1 * r n-1 Where: an is the nth term in the sequence a1 is the first term n is the number of the term r is the common ratio • Geometric Mean Find the product of the two values and then take the square root of the answer.

  5. Let’s start with the geometric mean • Find the geometric mean between 3 and 48 Let’s try one: Find the geometric mean between 28 and 5103

  6. Ex: Write the explicit formula for each sequence First term: a1 = 7 Common ratio = 1/3 Explicit: an = a1 * r n-1 a1 = 7(1/3) (1-1) = 7 a2 = 7(1/3) (2-1) = 7/3 a3 = 7(1/3) (3-1) = 7/9 a4 = 7(1/3) (4-1) = 7/27 a5 = 7(1/3) (5-1) = 7/81 Now find the first five terms:

  7. Explicit Arithmetic Sequence Problem Find the 19th term in the sequence of 11,33,99,297 . . . an = a1 * r n-1 Start with the explicit sequence formula Find the common ratio between the values. Common ratio = 3 a19 = 11 (3) (19-1) Plug in known values a19 = 11(3)18 =4,261,626,379 Simplify

  8. Let’s try one Find the 10th term in the sequence of 1, -6, 36, -216 . . . an = a1 * r n-1 Start with the explicit sequence formula Find the common ratio between the values. Common ratio = -6 a10 = 1 (-6) (10-1) Plug in known values a10 = 1(-6)9 = -10,077,696 Simplify

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