1. Geometric Sequences
Skill 50

2. Objective
HSF-BF.2: Write geometric sequences both recursively and with an explicit formula, use them to model situations and translate between two forms.

3. The Geometric Sequences
Starting Value = a
Common Ratio = r
𝑎, 𝑎∙𝑟, 𝑎∙𝑟∙𝑟, 𝑎∙𝑟∙𝑟∙𝑟,…
𝑎, 𝑎𝑟, 𝑎 𝑟 2 , 𝑎 𝑟 3 ,…
Recursive Formula:
𝑎 1 =𝑎
𝑎 𝑘 = 𝑎 𝑘−1 ∙𝑟
𝑎 𝑛 =…
Explicit Formula:
𝑎 𝑛 = 𝑎 1 ∙ 𝑟 𝑛−1

4. Example; Geometric Sequence
Is the sequence Geometric or Arithmetic?
2, 4, 6, 8, 10,… +2 +2 +2 +2
The sequence is Arithmetic because there is a common difference, not a common ratio.
5, −5, 5, −5, 5,…
x(-1)
x(-1)
x(-1)
x(-1)
The sequence is Geometric because there is a common ratio (-1) between terms.

5. Example; Geometric Sequences
Find a recursive and explicit formula.
7, 21, 63, 189,…
𝑟= 21 7 =𝟑
Recursive Formula.
𝑎 1 =…
𝑎 𝑘 = 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 …
Explicit Formula.
𝒂 𝟏 =𝟕
𝑎 𝑛 =…
𝒂 𝒌 = 𝒂 𝒌−𝟏 ∙𝟑
𝑎 𝑛 = 𝑎 1 ∙ 𝑟 𝑛−1
𝒂 𝒏 =𝟕 𝟑 𝒏−𝟏

6. Example; Geometric Sequences
Two managers at a clothing store created sequences to show the original price and the marked-down prices of an item. Write an explicit formula for each sequence. Find the price of the item be after the 6th markdown.
$60, $51, $43, $36.85,…
$60, $52, $44, $36,…
𝑟= =$𝟎.𝟖𝟓
𝑑=52−60=−$𝟖
𝑎 𝑛 = 𝑎 1 +𝑑 𝑛−1
𝑎 𝑛 = 𝑎 1 ∙ 𝑟 𝑛−1
𝑎 𝑛 =60−8 𝑛−1
𝒂 𝒏 =𝟔𝟎 𝟎.𝟖𝟓 𝒏−𝟏
𝑎 𝑛 =60−8𝑛+8
𝒂 𝒏 =−𝟖𝒏+𝟔𝟖
𝑎 7 = −1
𝑎 7 =−
𝒂 𝟕 =$𝟐𝟐.𝟔𝟑
𝒂 𝟕 =$𝟏𝟐.𝟎𝟎

7. #50: Geometric Sequences
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