Introduction to Geometric Progression - with examples

I want to introduce a fresh topic in math, one that has application in physics, engineering and other specialization areas that involve complex mathematics. It is Geometric Progression - also called Geometric sequence. We will learn what a Geometric Progression is and see some worked examples. I will also demonstrate how to find the nth term of a Geometric progression.

Understanding Geometric Progressions

A geometric progression is a type of sequence in which the common ratio is multiplied by the previous term to obtain the next term. The common ratio or constant factor determines the next and subsequent terms by doing a simple multiplication. Below are some examples of Geometric Progression:

• 4,12,36,108,324 (The common factor is x3 to get the next term)
• 7,14,28,56 (The common factor is x2 to get the next term)
• 1,5,25,125,625 (The common factor is x5 to get the next term)

As you can see with the above examples, the common factor in each is constant. And to get the next term, the common factor is multiplied by the previous term. These are Geometric progressions.

Finding the nth term of a Geometric Progression

For the simple examples above, its not difficult to look at the sequence and identify the common factor and the next terms. But for more complex examples, its that easy. So in order to find the nth term of a Geometric progression, you have to apply a formula. Below is the formula:

In the above formula:

a = the first term
n = Number of terms
r = the common ratio

We can apply this formula to solve for the nth term of geometric progressions. Now lets look at the following examples.

Solving for the nth Term

Example 1: Find the 12th term of the following 4,-8,16, -32

Solution:
T_n=ar^(n-1)

n = 12
a = 4
r = -8/4 0r 16/-8 = -2

Therefore,

T₁₂= 4 x (-2)^(12-1)
T₁₂= 4 x (-2)⁽¹¹⁾
T₁₂= 4 x -2048
T₁₂= -8,192

Example 2: Find the 7th term of 6, -18, 54.

Solution:
T_n=ar^(n-1)

n = 7
a = 6
r = -18/6 0r 54/-18 = -3

Therefore,

T₆= 6 x (-3)^(7-1)
T₆= 6 x (-3)⁽⁶⁾
T₆= 6 x -729
T₆= -4,374

Example 3: Find the 6th term of 4,-2,1

Solution:
T_n=ar^(n-1)

n = 6
a = 4
r = -2/4 0r 1/-2 = -1/2

Therefore,

T₆= 4 x (-1/2)^(7-1)
T₆= 4 x (-1/2)⁽⁶⁾
T₆= 4 x (-1/64)
T₆= -4/64
T₆= -1/16

Conclusion

These are pretty easy examples of Solving for the nth term of a geometric Progression. We have seen the formula and how to apply it. In the next article, we will looking at a more complex example where some of the terms of the progression are completely unknown.