**Geometric sequences**

**Geometric sequences**

Table of Contents

**Geometric sequence****A geometric sequence is a sequence of numbers in which each new term (except for the first term) is calculated by multiplying the previous term by a constant value called the constant ratio (r).**

**This means that the ratio between consecutive numbers in a geometric sequence is a constant (positive or negative). We will explain what we mean by ratio after looking at the following example.**

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**Example: A flu epidemic (EMCDS)**

**Example: A flu epidemic (EMCDS)**

**Influenza (commonly called “flu”) is caused by the influenza virus, which infects the respiratory tract (nose, throat, lungs). It can cause mild to severe illness that most of us get during winter time. The influenza virus is spread from person to person in respiratory droplets of coughs and sneezes. This is called “droplet spread”. **

**This can happen when droplets from a cough or sneeze of an infected person are propelled through the air and deposited on the mouth or nose of people nearby. It is good practice to cover your mouth when you cough or sneeze so as not to infect others around you when you have the flu. Regular hand washing is an effective way to prevent the spread of infection and illness.**

**Assume that you have the flu virus, and you forgot to cover your mouth when two friends came to visit while you were sick in bed. They leave, and the next day they also have the flu. Let’s assume that each friend in turn spreads the virus to two of their friends by the same droplet spread the following day. Assuming this pattern continues and each sick person infects 2 other friends, we can represent these events in the following manner:**

**Each person infects two more people with the flu virus.**

**We can tabulate the events and formulate an equation for the general case:**

Day (n) |
No. of newly-infected people |

1 |

2=2 |

2 |

4=2×2=2×21 |

3 |

8=2×4=2×2×2=2×22 |

4 |

16=2×8=2×2×2×2=2×23 |

5 |

32=2×16=2×2×2×2×2=2×24 |

⋮ |

⋮ |

n |

2×2×2×2×⋯×2=2×2n−1 |

**The above table represents the number of newly-infected people after n**

**days since you first infected your 2**

**friends.**

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**You sneeze and the virus is carried over to 2**

**people who start the chain (a=2). The next day, each one then infects 2 of their friends. Now 4 people are newly-infected. Each of them infects 2 people the third day, and 8**

**new people are infected, and so on. These events can be written as a geometric sequence:**

**2;4;8;16;32;…****Note the constant ratio (r=2**

**) between the events. Recall from the linear arithmetic sequence how the common difference between terms was established. In the geometric sequence we can determine the constant ratio (r**

**) from:**

**T2T1=T3T2=r****More generally,**

**TnTn−1=r****The general term for a geometric sequence (EMCDT)**

**The general term for a geometric sequence (EMCDT)**

**From the flu example above we know that T1=2**

**and r=2, and we have seen from the table that the nth term is given by Tn=2×2n−1**

**.**

**The general geometric sequence can be expressed as:**

**T1T2T3T4Tn=a=a×r=a×r×r=a×r×r×r=a×[r×r…(n−1) times]=ar0=ar1=ar2=ar3=arn−1****Therefore the general formula for a geometric sequence is:**

**Tn=arn−1****where**

**a****is the first term in the sequence;****r**

**is the constant ratio.**

**Test for a geometric sequence**

**To test whether a sequence is a geometric sequence or not, check if the ratio between any two consecutive terms is constant:**

**T2T1=T3T2=TnTn−1=r****If this condition does not hold, then the sequence is not a geometric sequence.**

**Worked example 3: Flu epidemic**

**Worked example 3: Flu epidemic**

**We continue with the previous flu example, where Tn**

**is the number of newly-infected people after n****days:**

**Tn=2×2n−1****Calculate how many newly-infected people there are on the tenth day.****On which day will 16 384**

**people be newly-infected?**

**Write down the known values and the general formula**

**Write down the known values and the general formula**

**arTn=2=2=2×2n−1****Use the general formula to calculate T10**

**Use the general formula to calculate T10**

**Substitute n=10**

**into the general formula:**

**Tn∴T10=a×rn−1=2×210−1=2×29=2×512=1024**

**On the tenth day, there are 1 024**

**newly-infected people.**

**Use the general formula to calculate n**

**Use the general formula to calculate n**

**We know that Tn=16 384**

**and can use the general formula to calculate the corresponding value of n****:**

**Tn16 38416 38428 192We can write 8 192So 213∴13∴n=arn−1=2×2n−1=2n−1=2n−1 as 213=2n−1=n−1(same bases)=14****There are 16 384**

**newly-infected people on the 14th day.**