Arithmetic & Geometric sequences, recursive formulae

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 assumethat they in turn spread 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 representthese events in the following manner:

Again we can tabulate the events and formulate an equation for the general case:

The above table represents the number of newly-infected people after $n$ days since you first infected your 2 friends.

You sneeze and the virus is carried over to 2 people who start the chain ( $a_1=2$ ). 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 people are infected, and so on. These events can be written as a geometric sequence:

Note the common factor (2) between the events. Recall from the linear arithmetic sequence how the common difference between terms were established. In the geometric sequence we can determine the common ratio , $r$ , by

Or, more general,

Common factor of geometric sequence :

Determine the common factor for the following geometric sequences:

1. $5,10,20,40,80,...$
2. $\frac{1}{2},\frac{1}{4},\frac{1}{8},...$
3. $7,28,112,448,...$
4. $2,6,18,54,...$
5. $-3,30,-300,3000,...$

General equation for the $n^{th}$ -term of a geometric sequence

From the above example we know $a_1=2$ and $r=2$ , and we have seen from the table that the $n^{th}$ -term is given by $a_n=2\times 2^{n-1}$ . Thus, in general,

where $a_1$ is the first term and $r$ is called the common ratio .

So, if we want to know how many people are newly-infected after 10 days, we need to work out $a_{10}$ :

That is, after 10 days, there are 1 024 newly-infected people.

Or, how many days would pass before 16 384 people become newly infected with the flu virus?

That is, 14 days pass before 16 384 people are newly-infected.

General equation of geometric sequence :

Determine the formula for the following geometric sequences:

1. $5,10,20,40,80,...$
2. $\frac{1}{2},\frac{1}{4},\frac{1}{8},...$
3. $7,28,112,448,...$
4. $2,6,18,54,...$
5. $-3,30,-300,3000,...$

Exercises

1. What is the important characteristic of an arithmetic sequence?
2. Write down how you would go about finding the formula for the $n^{\mathrm{th}}$ term of an arithmetic sequence?
3. A single square is made from 4 matchsticks. Two squares in a row needs 7 matchsticks and 3 squares in a row needs 10 matchsticks. Determine:
   1. the first term
   2. the common difference
   3. the formula for the general term
   4. how many matchsticks are in a row of 25 squares

   

4. $5;x;y$ is an arithmetic sequence and $x;y;81$ is a geometric sequence. All terms in the sequences are integers. Calculate the values of $x$ and $y$ .

Recursive formulae for sequences

When discussing arithmetic and quadratic sequences, we noticed that the difference between two consecutive terms in the sequence could be written in a general way.

For an arithmetic sequence, where a new term is calculated by taking the previous term and adding a constant value, $d$ :

The above equation is an example of a recursive equation since we can calculate the $n^{th}$ -term only by considering the previous term in the sequence. Compare this with equation [link] ,

where one can directly calculate the $n^{th}$ -term of an arithmetic sequence without knowing previous terms.

For quadratic sequences, we noticed the difference between consecutive terms is given by [link] :

Therefore, we re-write the equation as

which is then a recursive equation for a quadratic sequence with common second difference, $D$ .

Using [link] , the recursive equation for a geometric sequence is:

Recursive equations are extremely powerful: you can work out every term in the series just by knowing previous terms. As you can see from the examples above, working out $a_n$ using the previous term $a_{n-1}$ can be a much simpler computation than working out $a_n$ from scratch using a general formula. This means that using a recursive formula when using a computer to work out a sequencewould mean the computer would finish its calculations significantly quicker.

Recursive formula :

Write the first 5 terms of the following sequences, given their recursive formulae:

1. $a_n=2a_{n-1}+3,a_1=1$
2. $a_n=a_{n-1},a_1=11$
3. $a_n=2a_{n-1}^2,a_1=2$

The Fibonacci Sequence

Consider the following sequence:

The above sequence is called the Fibonacci sequence . Each new term is calculated by adding the previous two terms. Hence, we can write down the recursive equation: