Geometric Sequences

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Geometric Sequences
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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.

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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

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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.
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Lets start with the geometric mean

• Find the geometric mean between 3 and 48

Lets try one Find the geometric mean between 28
and 5103
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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
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Explicit Geometric Sequence Problem

• Find the 19th term in the sequence of
  11,33,99,297 . . .

Start with the explicit sequence formula
an a1 r n-1
Find the common ratio between the
values.
Common ratio 3
a19 11 (3) (19-1)
Plug in known values
a19 11(3)18 4,261,625,379
Simplify
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Lets try one

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

Start with the explicit sequence formula
an a1 r n-1
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