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
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:
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
Find the sum of the first 10 terms of the following Geometric Series
The sum of a geometric series can be found using the following formula:
From the above formula, we can see that we only need the first term, the common ratio and the number of terms in the series for us to calculate the sum. We already have the first term and the number of terms so lets proceed to find the common ratio.
Substituting in the formula gives the following: