Geometric Progression

A progression is another way of saying sequence thus a Geometric Progression is
also known as a Geometric Sequence.

A Geometric Progression is a special sequence defined by the special property that
the ratio of two consecutive terms is the same for all the terms in the sequence.
Whereas in Arithmetic Progression we talked of difference, here we talk of ratios
meaning that when you divide the current term by the previous term the number that
you get should be a non zero constant that is the same for all the consecutive pairs
of terms in the sequence. This number is known as the common ratio and is denoted
by the letter r

For example, given the sequence below

for the sequence to qualify as a geometric progression, the following should be
true

therefore since we have established the above relationship, we conclude that the
following should be true

and also for the 4th term

and the last term

So have you noticed anything yet?

The expression for last term gives us a general expression for finding any term
in a geometric progression. All you need is the first term and the common ratio
then apply the following:

For example; find the 5th and 8th terms in the geometric progression given that
the first term is 2 and the common ratio is 3

Solution:

Since we have the first term and the common ratio r, all we need to do is
substitute in the formula to obtain the terms we need:

Geometric Series

Since we have a geometric sequence, you should also expect to have a geometric series
for the sum of the terms in a geometric sequence.

Using the series notation, a geometric series can be represented as

Similar to what we did in Arithmetic Progression, we can derive a formula for finding
sum of a geometric series.

The first step is to substitute for the different terms and put the whole expression
in terms of only the first term

since the first term is common to the entire expression, we can factor it out as
follows:

If we were to multiply the sum by the common ratio r we would obtain the
following

Then we proceed to subtract the above from the original expression of the sum as
follows

which after some very long and tedious manipulation that you shouldn’t be too worried
about gives

factoring out the sum

which leaves the following as the formula for finding the sum of a geometric series

which is also expressed as

Examples of Geometric Progression