Geometric Progression, Series & Sums
Introduction
A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-1.jpg)
where | r | common ratio |
a1 | first term | |
a2 | second term | |
a3 | third term | |
an-1 | the term before the n th term | |
an | the n th term |
The geometric sequence is sometimes called the geometric progression or GP, for short.
For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence.
Note that after the first term, the next term is obtained by multiplying
the preceding element by 3.
The geometric sequence has its sequence formation:
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-2.jpg)
To find the nth term of a geometric sequence we use the formula:
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-3.jpg)
where | r | common ratio |
a1 | first term | |
an-1 | the term before the n th term | |
n | number of terms |
Sum of Terms in a Geometric Progression
Finding the sum of terms in a geometric progression is easily obtained by applying the formulas:
nth partial sum of a geometric sequence
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-6.jpg)
sum to infinity
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-7.jpg)
where | Sn | sum of GP with n terms |
S∞ | sum of GP with infinitely many terms | |
a1 | the first term | |
r | common ratio | |
n | number of terms |
Examples of Common Problems to Solve
Write down a specific term in a Geometric Progression
Question
Write down the 8th term in the Geometric Progression 1, 3, 9, ...
Answer
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-4.jpg)
Finding the number of terms in a Geometric Progression
Question
Find the number of terms in the geometric progression 6, 12, 24, ..., 1536
Answer
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-5.jpg)
Finding the sum of a Geometric Series
Question
Find the sum of each of the geometric series | ![]() |
Answer
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-9.jpg)
Finding the sum of a Geometric Series to Infinity
Question
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-10.jpg)
Answer
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-11.jpg)
Converting a Recurring Decimal to a Fraction
Decimals that occurs in repetition infinitely or are repeated in period are called recurring decimals.
For example, 0.22222222... is a recurring decimal because the number 2 is repeated infinitely.
The recurring decimal 0.22222222... can be written as
.
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-12.jpg)
Another example is 0.234523452345... is a recurring decimal because the number 2345 is repeated periodically.
Thus, it can be written as
or it can also be expressed in fractions.
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-13.jpg)
Question
Express
as a fraction in their lowest terms.
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-14.jpg)
Answer
![](http://mathematics.laerd.com/maths/img/geometricseries/geometric-series-15.jpg)
Try this question :
Question 1
![](http://mathematics.laerd.com/maths/img/geometricseries/question1.jpg)
Question 2
![](http://mathematics.laerd.com/maths/img/geometricseries/question2.jpg)
Question 3
![](http://mathematics.laerd.com/maths/img/geometricseries/question3.jpg)
Question 4
![](http://mathematics.laerd.com/maths/img/geometricseries/question4.jpg)
Question 5
![](http://mathematics.laerd.com/maths/img/geometricseries/question5.jpg)
Question 6
![](http://mathematics.laerd.com/maths/img/geometricseries/question6.jpg)
Question 7
![](http://mathematics.laerd.com/maths/img/geometricseries/question7.jpg)
Question 8
![](http://mathematics.laerd.com/maths/img/geometricseries/question8.jpg)
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