CHAPTER 3. SEQUENCES AND SERIES 3.3
You sneeze and the virus is carried over to 2 people who start the chain (a 1 = 2). The next day, each
one then infects 2 of their friends. Now 4 people are newly-infected. Each of them infects 2 people the
third day, and 8 people are infected, and so on. These events can be written as a geometric sequence:
2; 4; 8; 16; 32; ...
Note the common ratio( 2 ) between the events. Recall from the linear arithmetic sequence howthe
common difference between terms was established. In the geometric sequence we can determinethe
common ratio, r, from
a 2
a 1=
a 3
a 2
= r (3.4)Or, more generally,
an
an− 1
= r (3.5)
Activity: Common Ratio of Geometric Sequence
Determine the commonratio for the following geometric sequences:- 5; 10; 20; 40; 80;...
2.^12 ;^14 ;^18 ;... - 7; 28; 112; 448;...
- 2; 6; 18; 54;...
5.−3; 30;− 300; 3000;...
General Equation for the n
th
-Term of a Ge-
ometric SequenceEMCS
From the flu example above we know that a 1 = 2 and r = 2, and we have seen fromthe table that the
nth-term is given by an= 2× 2 n−^1. Thus, in general,
an= a 1 .rn−^1 (3.6)where a 1 is the first term and r is called the common ratio.
So, if we want to knowhow many people are newly-infected after 10 days, we need to work out a 10 :
an = a 1 .rn−^1
a 10 = 2× 210 −^1
= 2× 29
= 2× 512
= 1024That is, after 10 days, there are 1 024 newly-infected people.