Geometric Sequences Skill 50

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

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

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.

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

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= 51 60 =$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 =60 .85 7−1 𝑎 7 =−8 7 +68 𝒂 𝟕 =$𝟐𝟐.𝟔𝟑 𝒂 𝟕 =$𝟏𝟐.𝟎𝟎

#50: Geometric Sequences Questions Summarize Notes Homework Google Classroom Quiz