1

––

N

1

––

1 – r

1

––

1 – r

1

––

2

1

––

8

1

––

4

1

––

2

geometric sequence 223

Associativity. This rule holds for all matrices, and

hence for invertible matrices in particular.

Inverses. The inverse of a matrix Ais itself invertible,

and so belongs to the set: (A–1)–1 = A.

A study of

DETERMINANT

s and C

RAMER

’

SRULE

shows that a matrix is invertible if, and only if, its

determinant is not zero. Thus the nth-order general lin-

ear group may be defined as the set of all n×nsquare

matrices with nonzero determinants. The subset of all

square matrices with determinant equal to 1 forms a

subgroup of GLncalled the special linear group,

denoted SLn.

geodesic The shortest curve connecting two points

on a surface, and lying wholly on that surface, is called

a geodesic. For example, P

YTHAGORAS

’

S THEOREM

shows that a straight line gives the shortest path

between two points on a plane, and thus straight lines

are the geodesics of a plane. On a

SPHERE

, geodesics are

sections of great circles. (Planes fly along geodesic arcs

across the Earth’s surface.) One envisions a geodesic as

the path a band of stretched elastic would adopt if held

at the two points in question and forced to remain on

the surface under study.

On a plane there is always only one geodesic

between any two given points. On a sphere there are

infinitely many geodesics that connect the two poles of

the sphere, for example. (If the two points chosen are

not antipodal, however, then the geodesic connecting

them is unique.)

A geodesic dome is a domelike structure made of

straight-line structural elements held in tension. The

straight line segments approximate geodesics of the

dome.

geometric distribution See

BINOMIAL DISTRIBUTION

.

geometric mean See

MEAN

.

geometric sequence (geometric progression) A

SE

-

QUENCE

of numbers in which each term, except the

first, is a fixed multiple of the previous one is called a

geometric sequence. The constant ratio of terms is

called the common ratio. For example, the sequence

1,3,9,27,… is geometric, with common ratio 3. The

sequence 1, , , ,... is also geometric, with common

ratio .

A geometric sequence with first term aand com-

mon ratio rhas the form: a, ar, ar2, ar3,… The nth term

anof the sequence is given by an= arn–1. (It is common,

however, to start the count by calling the first term of

the sequence the “zero-th term,” so that a0= ar0= a

and the nth term of the sequence is given by the for-

mula: an= arn.) A geometric sequence can also be

described as exponential. The common ratio ris the

base of the

EXPONENTIAL FUNCTION

f(n) = arn.

If the value rlies between –1 and 1, then the terms

arnof geometric sequence approach the value 0 as n

becomes large. If r= 1, then the geometric sequence is

the constant sequence a,a,a,… For all other values of r,

the sequence diverges.

The sum of the terms of a geometric sequence is

called a geometric series:

a+ ar + ar2+ ar3+…

In the study of

CONVERGENT SERIES

, the ratio test shows

that this series sums to a finite value if –1 < r< 1. The

value Sof the sum can be computed as follows:

If S= a+ ar + ar2+ ar3+…, then rS = ar + ar2+ ar3+ ar4

+… Subtracting yields: S– rS = a, and so S=

In particular we have 1 + r +r2+r3+ … = .

If ris a fraction of the form (with N≥2), then this

formula can be rewritten:

This particular expression can also be justified with a

physical demonstration, which we illustrate here for

the case N= 3:

John takes a piece of paper and tears it into

thirds. He hands one piece to Andrea, another