Geometric Sequence Skill 38

Objectives… Recognize, write, and find the nth terms of geometric sequences. Find nth partial sums of geometric sequences. Find sums of infinite geometric series. Use geometric sequences to model and solve real-life problems.

We know that a sequence whose consecutive terms have a common difference is an arithmetic sequence. In this section, you will study another important type of sequence called a geometric sequence. Consecutive terms of a geometric sequence have a common ratio.

Example-Geometric Series a. Find the common ratio in the geometric series. 𝟑, 𝟗, 𝟐𝟕, 𝟖𝟏, 𝟐𝟒𝟑, …, 𝟑 𝒏 , … Common ratio = 3 9 3 =3 27 9 =3 81 27 =3

Example-Geometric Series b. Find the common ratio in the geometric series. 𝟏𝟎, 𝟐𝟎, 𝟒𝟎, 𝟖𝟎, 𝟏𝟔𝟎, …, 𝟓 𝟐 𝒏 , … Common ratio = 2 20 10 =2 40 20 =2 80 40 =2

Example-Geometric Series c. Find the common ratio in the geometric series. − 𝟏 𝟒 , 𝟏 𝟏𝟔 , − 𝟏 𝟔𝟒 , 𝟏 𝟐𝟓𝟔 , − 𝟏 𝟏𝟎𝟐𝟒 , …, − 𝟏 𝟒 𝒏 , … Common ratio = − 𝟏 𝟒 1 16 −1 4 =− 1 4 −1 64 1 16 =− 1 4 1 256 −1 64 =− 1 4

𝒂 𝒏 =𝟑 𝟐 𝒏−𝟏 𝒂 𝒏 =𝟐 𝟒 𝒏−𝟏 a) First Term = 3; Common Ratio = 2 𝒂 𝒏 =𝟑 𝟐 𝒏−𝟏 b) First Term = 2; Common Ratio = 4 𝒂 𝒏 =𝟐 𝟒 𝒏−𝟏

Example-Geometric Sequences 𝒂 𝟏 =𝟐𝟎 and 𝒓=𝟏.𝟎𝟓, find 𝒂 𝟏𝟓 = ? 𝑎 𝑛 = 𝑎 1 𝑟 𝑛−1 𝑎 𝑛 =20 1.05 𝑛−1 𝒂 𝟏 =𝟒 and 𝒓= 𝟏 𝟐 , find 𝒂 𝟗 = ? 𝑎 15 =20 1.05 15−1 𝒂 𝟏𝟓 =𝟑𝟗.𝟔𝟎 𝑎 𝑛 = 𝑎 1 𝑟 𝑛−1 𝑎 𝑛 =4 1 2 𝑛−1 𝑎 9 =4 1 2 9−1 𝒂 𝟗 =.𝟎𝟏𝟓𝟔𝟐𝟓

Example-Geometric Sequences 𝒂 𝟒 =𝟏𝟐𝟓 and 𝐚 𝟏𝟎 = 𝟏𝟐𝟓 𝟔𝟒 , find 𝒂 𝟏𝟒 = ? See Video over Geometric Series 𝒂 𝟐 =−𝟏𝟖 and 𝐚 𝟓 = 𝟐 𝟑 , find 𝒂 𝟔 = ? See Video over Geometric Series

The formula for the sum of a finite geometric sequence is as follows.

Example-Find the Sum 𝑎 1 =12 𝑟=3 𝑎 1 =1 𝑟=2 𝑆 12 =12 1− 3 12 1−3 𝑖=1 12 4 3 𝑖 𝑖=1 7 2 𝑖−1 𝑎 1 =12 𝑟=3 𝑎 1 =1 𝑟=2 𝑆 12 =12 1− 3 12 1−3 𝑆 7 =1 1− 2 7 1−2 𝑆 12 =12 265720 𝑺 𝟕 =𝟏𝟐𝟕 𝑺 𝟏𝟐 =𝟑,𝟏𝟖𝟖,𝟔𝟒𝟎

Example-Find the Sum 𝑎 1 =4 𝑟=.6 𝑎 1 =5 𝑟=.5 𝑆 ∞ = 4 1−.6 𝑆 ∞ = 5 1−.5 𝒊=𝟏 ∞ 𝟒 .𝟔 𝒊 𝒊=𝟏 ∞ 𝟓 𝟏 𝟐 𝒊 𝑎 1 =4 𝑟=.6 𝑎 1 =5 𝑟=.5 𝑆 ∞ = 4 1−.6 𝑆 ∞ = 5 1−.5 𝑆 ∞ = 4 .4 𝑺 ∞ =𝟏𝟎 𝟑+.𝟑+.𝟎𝟑+.𝟎𝟎𝟑+… 𝑺 ∞ =𝟏𝟎 𝑺 ∞ = 𝟏𝟎 𝟑

38 Geometric Sequences Summarize Notes Videos Homework Worksheet Quiz