     Home Arithmetic Sequences Geometric Sequences Arithmetic Series Geometric Series Sigma Notation Mathematical Induction Review&Test 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 , t3 , 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   tn = arn-1.

·        Geometric sequences are exponential functions with domain the natural numbers N = {1, 2, 3, 4, …} Definition: The sum of the terms of a Geometric sequence is a Geometric Series.  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 tn.

 a = 2 r = 4 n = ? tn = 32768 Sn = ? 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:    