The Art and Craft of Problem Solving

158 CHAPTER 5 ALGEBRA

Upon adding, we immediate deduce that

the intuitively reasonable fact that the sum is equal to the average value of the terms

multiplied by the number of terms. It is no coincidence that another term for "average"

is arithmetic mean.

Geometric Series and the Telescope Tool

A geometric sequence is exactly like an arithmetic sequence except that now the

consecutive terms have a common ratio; i.e., the sequence has the form

a,ar,ar^2 ,ar ,^3

....

The Gaussian pairing tool is no help for summing geometric series, because the terms

are not additively symmetric. However, the wonderful telescope tool comes to the

rescue. Consider a geometric series of n terms with first term a and common ratio r

(so the last term is arn-l). Rather than write the sum S twice, we look at S and rS:

S = a + ar + ar^2 + ... + arn-l ,

rS = ar+ar^2 +ar^3 + ... +arn.

Observe that Sand rS are nearly identical, and hence subtracting the two quantities

will produce something very simple. Indeed,

S -rS = a -ar+ar -ar^2 +ar^2 - ar^3 + ... +arn-

l

• arn,
  and all terms cancel except for the first and the last. (That's why it's called "telescop­
  ing," because the expression "contracts" the way some telescopes do.) We have

and solving for S yields

S -rS = a -arn ,

a-arn

S--­

• l - r
  .

Geometric series crop up so frequently that it is probably worth memorizing this for­

mula. In any event, the crux move-the telescope tool-must be mastered.

There are many ways to telescope a series. With the geometric series above, we

created two series that were virtually the same. The next series, one that you first saw

as Example 1.1. 2 on page 1, requires different treatment.

Example 5.3.1 Write

1 1 1 1

1·2

+

2· 3

+

3· 4

+ ... +

99 · 100

as a fraction in lowest terms.

Solution: Notice that each term can be written as

1 1 1

k(k+ 1) k