LESSON 1

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UNIT 10 : SEQUENCES AND SERIES

LESSON 4: GEOMETRIC SERIES

Geometric Series:

Recall a sequence such as 2, 4, 8, 16, 32, … is called a Geometric Sequence. These sequences have the following properties.

· Terms are denoted as t₁ , t₂ , t_{3 ,}referring to term1, term 2, term 3 …

· This ratio is called the common ratio and denoted using the letter r. Here r = 2.

· The first term is denoted using the letter a. Here a = 2.

· Successive terms are found by multiplying a given term by the common ratio. Eg. t₆ = 32 x 2 = 64 etc.

· The formula for the general term or nth term is t_n= ar^n-1.

· Geometric sequences are exponential functions with domain the natural numbers N = {1, 2, 3, 4, …}

Text Box: a = t1 = first term
r = common ratio
n = term number

Definition: The sum of the terms of a Geometric sequence is a Geometric Series.

Text Box: n = the number of terms
t1 = a = the first term
r = the common ratio
Sn = the sum of the series

Text Box:

Example 1:

Form the geometric sequence determined by the exponential function f(n) = 3(2)^n-1. Find a, r and S₁₀.

Solution:

a = 3

r = 2

n = 10

S₁₀ = ?

a = - 2

r = 3

n = 9

S₉ = ?

Example 2: Given the first few terms.

Given the series –2 – 6 – 18 – 54 – …

a) Show that the series is geometric.

b) Find S₉

Solution:

Example 3: Given the first and last terms.

Given the series 2 + 8 + 32 + … + 32768. Find the sum of the series.

Solution: First find the number of terms n.

Let the last term be t_n.

a = 2

r = 4

n = ?

t_n = 32768

S_n = ?

Example 4: When r is negative or fractional.

a) Given the geometric series –3 + 6 – 12 + 24 – … Find a, r and S₁₂.

Solution:

Infinite Geometric Series: