Let's learn about the geometric sequence with some examples in A-Level Maths!

Geometric Sequences 

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

In General we  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") 

But be careful, r should not be 0:   

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

Example:    {1,2,4,8,...} 

The sequence starts at 1 and doubles each time, so  

• a=1 (the first term)  
• r=2 (the  "common ratio" between terms is a doubling)  

And we get: 

{a, ar, ar², ar³, ... } 

= {1, 1×2, 1×2², 1×2³, ... }  

= {1, 2, 4, 8, ... } 

The Rule 

We can also calculate any term using the Rule:

Example:   

 10, 30, 90, 270, 810, 2430,  ...    

This sequence has a factor of 3 between each number. 

The values of a and r are:  

• a = 10 (the first term)  
• r = 3 (the  "common ratio")  

The Rule for any term is: 

x_n = 10 × 3^(n-1) 

So, the 4th term is: 

x₄ = 10×3^(4-1) = 10×3³ = 10×27 = 270 

And the 10th term is: 

x₁₀ = 10×3^(10-1) = 10×3⁹ = 10×19683 = 196830 

A Geometric Sequence can also have smaller and smaller values:  

Example:   4, 2, 1, 0.5, 0.25, ...    

This sequence has a factor of 0.5 (a half) between each number.

Its Rule is x_n = 4 × (0.5)^n-1 

Summing a Geometric Series 

To sum these: 

a + ar + ar² + ... + ar^(n-1) 

(Each term is ar^k, where k starts at 0 and goes up to n-1)

Example: 

Sum the first 4 terms of  10, 30, 90, 270, 810, 2430,  ...    

This sequence has a factor of 3 between each number.   

The values of a, r and n are:  

• a = 10(the first term)  
• r = 3 (the  "common ratio") 
• n = 4 (we want to sum the first 4 terms) 

Example: Add up the first 10 terms of the Geometric Sequence that halves each time:  {  1/2, 1/4, 1/8, 1/16, ... } 

The values of a, r and n are:  

• a =  ½ (the first term) 
• r = ½ (halves each time) 
• n = 10 (10 terms to add) 

Infinite Geometric Series 

So what happens when n goes to  infinity?  

We can use this  formula: 

***r must be between (but not including) −1 and 1 and r should not be 0 because  the sequence {a,0,0,...}  is not geometric 

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Drafted by Eunice (Maths)

Reference

https://www.mathsisfun.com/algebra/sequences-sums-geometric.html