Geometric Sequences.

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Presentation transcript:

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. We look for multiplication to identify geometric sequences.

Ex: Determine if the sequence is geometric 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

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

Ex: Write the recursive formula for each sequence First term: a1 = 7 Common ratio = 1/3 Recursive: an = an-1 * r an = an-1 * (1/3) a1 = 7 a2 = 7(1/3) = 7/3 a3 = 7/3(1/3) = 7/9 a4 = 7/9(1/3) = 7/27 a5 = 7/27(1/3) = 7/81 Now find the first five terms:

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:

Recursive Geometic Sequence Problem Find the 5th and 6th term in the sequence of 11,33,99,297 . . . an = an-1 * r Start with the recursive sequence formula Find the common ratio between the values. Common ratio = 3 a5 = 297 (3) = 891 Plug in known values a6 = 891(3) = 2673 Simplify

Explicit Geometic 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

Find the 10th term in the sequence of 1, -6, 36, -216 . . . 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