Sequences


Introduction

Consider a sequence

1 2 3 4 5
5 8 13 20 29 ...

The top row indicates the number of the term within the sequence, while the second row is the sequence itself.

Now what we are saying is that there is a connection between each number on the top row and the number immediately below it. This relationship is indicated by a general formula of the type

3n+2,    4n2 + 1     or something similar

where n indicates the number on the top row. For example, I know that the formula for the above sequence is

n2 + 4

so for the first term, n=1 and the number below it will be 12 + 4 = 5. For the fourth term, n=4, and the number below it will be 42 + 4 = 20.

Representing the sequence by a formula produces

1. a compact representation
2. a representation which represents every single term, whereas originally the sequence was only represented by the first few terms, with the nature of its progression only being suggested.

These type of sequences commonly appear in the quiz and puzzles section of newspapers and magazines, where the answer can often be arrived at 'mentally' by trial and error. Here we will introduce a more systematic method.


First Differences and Second Differences


Consider the same sequence we considered in the previous section

1 2 3 4 5
5 8 13 20 29 ...

We can work out rows of 'differences'(see table below). The first differences are the differences between each term in the original sequences and this are presented on a row below the original sequence. If these first difference are not constant, we can do a similar operation by finding the differences between the terms in the first differences and likewise forming another row below the first differences - this row of numbers will be the second differences. If either the first difference or second differences are constant, then this is highly significant - as we show in the next sections.

5       13     20    29   ...
  3         9     ...
      2         ...

First Differences are Constant


If the first difference are constant, say d, then the general formula will be of the form

dn + k

i.e.

•  The formula will contain an n, with no index - this n will be the term number
•  The coefficient of this n will be equal to the value of the constant first diffference
•  To this will be added a k, a constant

To calculate this constant k, compare the general term with the original sequence.

Example

Consider the following sequence and first differences

5       11     14    17   ...
  3         3     ...

We can say immediately that the formula will be of the form

\[ 3n + k \]

Compare this with the original sequence.

For $n=1$ \[ 3n + k = 3+k\] The first term in the original sequence is $5$.
This implies that $k =2$

Check for $n=2$ \[ 3n + k = 6+k\] The second term in the original sequence is $8$.
This again implies that $k =2$

Inspection of other terms will show that $k$ does indeed equal $2$.

The required formula is

\[ 3n + 2 \]



Second Difference are Constant


If the second difference are constant, say d, then the general formula will be of the form

\[ \frac{d}{2} n^2 + fn + k \]

i.e.

•  The formula will contain an n2 - this n will be the term number
•  The coefficient of this n2 will be equal to half the value of the constant second diffference
•  To this will be added an n term (with a coefficient) and k, a constant

To calculate the latter two terms, compare the general term with the original sequence. If you are lucky the fn term will be absent and you will just need to find a constant. If however this comparison does not produce a constant value, you will need to operate on this new set of differences and operate on it as explained in the previous section (First Differences are Constant).

Example

Consider the following sequence and first and second differences

4       12     19    28   ...
  3         9     ...
      2         ...

We can say immediately that the formula will be of the form

n2 + fn + k

Although at GCSE level, it is more likely that required sequence will be of a simpler form where f = 0, i.e.

n2 + k
Compare this with the original sequence.

For n=1; n2 + k = 1 +k.
Note the first term in the original sequence is 4.

For n=2; n2 + k = 4 +k.
Note the second term in the original sequence is 7.

Since the difference between n2 and the original sequence is 3 in both these cases, this implies that k = 3.

Inspection of one or two other terms will show this to be the case.

So final required formula is

n2 + 3