**Introduction to Growth by Multiplication**

Another type of progression has values that are repeatedly multiplied by some constant. Here are a few examples:

*G* = 1, 2, 4, 8, 16, 32

*H* = 1, −1, 1, −1, 1, −1, . . .

*I* = 1, 10, 100, 1000

*J* = −5, −15, −45, −135, −405

*K* = 3, 9, 27, 81, 243, 729, 2187,. . .

*L* = ½, ¼, 1/8, 1/16, 1/32,. . .

Sequences *G* , *I* , and *J* are finite. Sequences *H* , *K* , and *L* are infinite, as indicated by the three dots following the last term in each sequence.

**Geometric Progression**

Examine the six sequences above. Upon casual observation, they appear to be much different from one another. But in all six sequences, each term is a specific and constant multiple of the term before it. Note:

- The values in
*G*progress by a constant factor of 2. - The values in
*H*progress by a constant factor of −1. - The values in
*I*progress by a constant factor of 10. - The values in
*J*progress by a constant factor of 3. - The values in
*K*progress by a constant factor of 3. - The values in
*L*progress by a constant factor of ½.

Each sequence has a starting point or first number. After that, succeeding numbers are generated by repeated multiplication by a constant. If the constant is positive, the values in the sequence stay “on the same side of 0” (they either remain positive or remain negative). If the constant is negative, the values in the sequence “alternate to either side of 0” (if a given term is positive, the next is negative, and if a given term is negative, the next is positive).

Let *t* _{0} be the first number in a sequence *T* , and let *k* be a constant. Imagine that *T* can be written in this form:

*T* = *t* _{0} , *t* _{0} *k* , *t* _{0} *k* ^{2} , *t* _{0} *k* ^{3} , *t* _{0} *k* ^{4} , . . .

for as long as the sequence goes. Such a sequence is called a *geometric sequence* or a *geometric progression* .

If *k* happens to be equal to 1, the sequence consists of the same number, listed over and over. If *k* = −1, the sequence alternates between *t* _{0} and its negative. If *t* _{0} is less than −1 or greater than 1, the values get farther and farther from 0. If *t* _{0} is between (but not including) −1 and 1, the values get closer and closer to 0. If *t* _{0} = 1 or *t* _{0} = −1, the values stay the same distance from 0.

The numbers *t* _{0} and *k* can be whole numbers, but this is not a requirement. As long as the multiplication factor between any two adjacent terms in a sequence is the same, the sequence is a geometric progression. In the last sequence, *k* = ½. This is an especially interesting sequence, as we’ll see in a moment.

**Geometric Series**

For a geometric sequence, the corresponding *geometric series,* which is the sum of all the terms, can always be defined if the sequence is finite. Sometimes the sum of all the terms can be defined even if the sequence is infinite.

For the above sequences *G* through *L* , let the corresponding series be called G _{+} through *L* _{+} . Then:

*G* _{+} = 1 + 2 + 4 + 8 + 16 + 32 = 63

*H* _{+} = 1 −1 + 1 −1 + 1 −1 + . . . = ?

*I* _{+} = 1 + 10 + 100 + 1000 = 1111

*J* _{+} = −5 −15 −45 −135 −405 = −605

*K* _{+} = “blows up” and is not defined

*L* _{+} = ½ + ¼ + 1/8 + 1/16 + 1/32 + . . . = ?

The finite series *G* _{+} , *I* _{+} , and *J* _{+} are straightforward enough. The infinite series *H* _{+} seems unable to settle on 0 or 1, repeatedly hitting both. It’s tempting to say that *H* _{+} is a number with two values at once, and a fascinating theory can be built around the notion of multi-valued numbers. But in conventional math, we have to say that *H* _{+} is not definable as a number. The infinite series *K* _{+} runs off “out of control” and is an example of a *divergent series,* because its values keep on getting farther and farther away from 0 without limit.

That leaves us with *L* _{+} . What’s going on with this series?

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