Question

Is the sequence geometric? 2, 6, 18, 54,...
A.) yes
B.) no

Answer 1

Sequence

A Sequence is a set of things (usually numbers) that are in order.

Answer 2

If you multiply /divide to get the next term, then it is a geometric sequence.

Answer 3

Geometric Sequences

In a Geometric Sequence each term is found by multiplying the previous term by a constant.

Answer 4

In General you could write a Geometric Sequence like this:

{a, ar, ar2, ar3, ... }

where:

a is the first term, and
r is the factor between the terms (called the "common ratio")

Answer 5

But be careful, r should not be 0:

When r=0, you get the sequence {a,0,0,...} which is not geometric

Answer 6

The Rule

You can also calculate any term using the Rule:

xn = ar(n-1)

(We use "n-1" because ar0 is for the 1st term)

Answer 7

A Geometric Sequence can also have smaller and smaller values:

Answer 8

Why "Geometric" Sequence?

Because it is like increasing the dimensions in geometry:

Answer 9

Summing a Geometric Series

When you need to sum a Geometric Sequence, there is a handy formula.

To sum:

a + ar + ar2 + ... + ar(n-1)

Each term is ark, where k starts at 0 and goes up to n-1

Use this formula:

Answer 10

The formula is easy to use ... just "plug in" the values of a, r and n