Sequences and Proof

When I started look­ing at “Sequences and Proof”, I fell into the trap of look­ing at some of the most dif­fi­cult ques­tions (A and A*) first. What a sur­prise to find that I couldn’t do them! I now think it’s espe­cially impor­tant with this topic, to mas­ter the basic, sim­ple stuff and then move onto more chal­leng­ing areas such as proof using algebra.

So, the basics:-

A sequence is a list of num­bers in a given order

Con­sec­u­tive num­bers are num­bers next to each other

We call the num­bers in a sequence, terms


This leads us to the most basic sequence:-

A liner sequence increases or decreases in equal sized steps.


For exam­ple:-   2, 4, 6, 8

In this sequence the terms increase by 2 in each step.

We can say:  The nth term = 2n (i.e the 1st term = 1 x 2 = 2, the 2nd term = 2 x 2 = 4, the 3rd term = 3 x 2 = 6 and the 4th term = 4 x 2 = 8).

When you’re try­ing to under­stand a sequence it’s very use­ful to write the dif­fer­ences between each term under the sequence, like this:-

Simple Sequence

 

 

 

 

When you have a lin­ear sequence if you look at dif­fer­ences between the terms to start to define the sequence, if the dif­fer­ences are all 2 then the sequence def­i­n­i­tion will include 2n, if the dif­fer­ences are all 3 then the sequence def­i­n­i­tion will include the term 3n, if the dif­fer­ences are all 4 then the the sequence def­i­n­i­tion will include the term 4n etc., etc.

This is best explained with a cou­ple of examples:-

Find the nth term of the sequence 4, 6, 8, 10,.…

First look for the dif­fer­ence between each term:-

Simple Linear Sequence

 

 

 

 

You now know that the sequence includes 2n. Set the infor­ma­tion in a table includ­ing 2n:-

2n sequence

 

 

 

 

 

 

 

 

 

This method might seem over the top for sim­ple lin­ear sequences, but it shows its worth with more com­pli­cated examples:-

Find the nth term of the sequence 1, 9, 17, 25

So start by find­ing the dif­fer­ence between each term and then com­plete the grid as shown above:-

Linear sequence example 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Most ques­tions ask you to find the nth term but some­times this is turned on its head. For exam­ple what is the 10th term if the nth term = 8n — 7? Just feed the actual num­ber into the statement:

10th term = (8 x 10) — 7 = 73 

Alter­na­tively you might be asked to find out if a par­tic­u­lar num­ber is a term in the sequence, for exam­ple is 155 a term in the above sequence? Again just feed the num­ber into the sequence definition.

If 155 is in the sequence, then 8n — 7 = 155

and 8n = 162

n = 162/8 = 20.25

So “n” is not a whole num­ber and 155 is not in the sequence.

Another exam­ple using the same sequence, is 177 a term in the sequence?

If 177 is in the sequence then 8n — 7 = 177

and 8n = 184

n = 184/8 = 23

So “n” is a whole num­ber and 177 is in the sequence.

Qua­dratic Sequences

The nth term of a qua­dratic includes an “n²”. To find out if a sequence is qua­dratic you check the sec­ond dif­fer­ences. So what is a sec­ond dif­fer­ence? We used the first dif­fer­ences between terms when we looked at lin­ear sequences above. The sec­ond dif­fer­ence is the dif­fer­ence between the dif­fer­ences. This is best under­stood by using an exam­ple. Let’s invent our own qua­dratic sequence, n² + 1. So look­ing at the first few terms and the dif­fer­ences and the sec­ond dif­fer­ences we have:-

quadratic sequence example

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

You may get a ques­tion where the answer is a mul­ti­ple or frac­tion of n². In these cases the sec­ond dif­fer­ence will not be 2, but it will be con­stant. For exam­ple, find the nth term of this sequence:-    0.25, 1, 2.25, 4, 6.25

Quadratic sequence example 0.25n2

 

 

 

 

 

 

 

 

 

 

 

 

 

Proofs and Proof using alge­bra to follow.

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One Response to Sequences and Proof

  1. sarim ali says:

    when doing the nth term with: 0.25, 1, 2.25, 4, 6.25 how did u work it out. i get it but how did it come to 0.25? because in the last one you took the sequence away from the nsquared but now you didn’t?

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