Transcript 9._Sequences

Maths Notes
Number
9. Sequences
www.mrbartonmaths.com
9. Sequences
Things you might need to be able to do with sequences…
This will vary with your age and what maths set you are in, but here is a list of some of
the things you might need to be able to do with sequences:
1.
2.
3.
4.
Spot and describe number sequences
Work out the nth term of linear sequences
Write out terms of sequences given the rule
Work out the nth term of quadratic sequences
It’s quite a nice little topic this one… well, as far as maths topics go, anyway!
What is a sequence?
A sequences is just a set of numbers which follows a rule.
The rule may be very simple, or very complicated, but the important thing is that every
single number in that sequence follows the same rule.
The reason this is important is that allows you to predict what number will come next, and
even what number will come in 1,000,000 numbers time!
1. Spotting and Describing Number Sequences
In these types of questions you will usually be given a set of numbers, and be asked to
describe the rule (how to you get from one number to the next), and predict what the
next couple of numbers will be.
Let’s have a look at some sequences together:
1.
7
10
13
16
19
…
…
2.
3
6
12
24
48
…
…
3.
200
190
181
173
166
…
…
4.
1
1
2
3
5
8
13
…
…
1. Here the numbers are going up by 3 every time, so the rule is something like: “add 3 to the
previous number to get the next number”, and so the next two numbers are 22 and 25
2. Here the numbers are doubling (or multiplying by 2), so the rule is something like: “double the
previous number to get the next number”, and so the next two numbers are 96 and 192
3. This one is a little tricky to spot, and even trickier to describe. I would go for something like:
“subtract 10 from the 1st number, 9 from the 2nd, 8 from the 3rd, and so on”. So long as you are
clear, you will get the marks. So, the next two numbers must be: 160 and 155
4. This is a sneaky one. It’s a very famous sequence called “The Fibonacci Sequence”. Here is
the rule: “add the previous two numbers together to get the next number”. That means what
must come next are: 21 and 34
2. Finding the nth term of Linear Sequences
Not the greatest sounding title in the world, hey?
Let’s have a look at what each bit means, and you’ll see it’s not so bad:
nth term - well, term is just a posh word for the numbers in a sequence, and n is just
the letter we use to describe the position of each term. So, n = 1 is the 1st term, and
n = 5 is the 5th term. All “find the nth term” means is just to find a rule which allows
us to work out what number lies at any position in our sequence.
linear sequences - these are just sequences where you either add or subtract the
same number to get from term to term. For example, sequence 1. up above was a
linear sequence because you added 3 each time, but number none of the other were
(in 2. we multiplied, and in 3. and 4. we added or subtracted a different amount).
Now I have a method for finding the nth term of linear sequences:
1. Decide what you have to add or subtract to get from term to term (and make
sure it is the same for each term!)
2. Write the times table of this number underneath the sequence (this gives you
the number that goes in front of n)
3. Figure out what you have to do to your times table to get back to your sequence
(this gives you the number at the end)
Example 1
Find the nth term of the following sequence:
13
19
25
31
…
37
…
1. Okay, now looking at the numbers I reckon you have to add 6 each time, but before I
continue, I am just going to check each term to make sure… erm… yep, add 6 each time!
2. Adding 6 each time means two things: (1) 6 is the number in front of n in my rule, so I know there
will be a 6n involved (2) I need to write the 6 times table carefully under my sequence:
n
Sequence:
6n
1
2
3
4
5
…
…
13
19
25
31
37
…
…
6
12
18
24
30
…
…
Notice: I have also written n above the sequence, just to remind you that all n means is the
position of the numbers in the sequence (n = 4 is the 4th number in the sequence, which is 31).
Also, notice how the 6 times table is just 6 times as big as n!
3. Now you ask yourself: “what do I have to do to get from my 6 times table, back to my
sequence?”… well, if you look carefully at the numbers, you will see you must… add 7 each time!
So, our rule for the nth term is…
6n  7
Which basically says that our sequence is just the 6 times table, with 7 added each time.
Notice: you can check you are correct by testing out your rule. We know the 5th term of the
sequence is 37, but does are rule give us that?...
When n = 5
6n + 7
6x5+7
=
… 37!
we are correct!
Example 2
Find the nth term of the following sequence:
-20
-16
-12
-8
…
-4
…
1. Okay, I know there are nasty negatives, but if you look carefully you should be able to see
that you have to add 4 each time.
2. Again, adding 4 means two things: (1) 4 is the number in front of n in my rule, so I know there will
be a 4n involved (2) I need to write the 4 times table carefully under my sequence:
n
Sequence:
4n
1
2
3
4
5
…
…
-20
-16
-12
-8
-4
…
…
4
8
12
16
20
…
…
Notice: Again, I have put n on the top just to show you that that 4 times table is just 4 times as
big as n!... that’s all 4n means, just get n and multiply it by 4!
3. Now you ask yourself: “what do I have to do to get from my 4 times table, back to my
sequence?”… well, again you have to be careful, but I reckon you must… erm… subtract 24 each time!
So, our rule for the nth term is…
4n  24
Important: as well as checking, we can also use this rule to predict. For example, we can very quickly
work out what the 100th term would be without writing out the whole sequence:
When n = 100
4n – 24
4 x 100 – 24
=
… 376!
Example 3
Find the nth term of the following sequence:
21
16
11
6
…
1
…
1. Okay, this time looking at the numbers I reckon you have to subtract 5 each time.
2. Now, just because we are subtracting, our method still works! Subtracting 5 still means two
things: (1) -5 is the number going in front of the n (2) we need a times table… the -5 times table!
n
Sequence:
-5n
1
2
3
4
5
…
…
21
16
11
6
1
…
…
-5
-10
-15
-20
-25
…
…
Notice: the -5 times table is just the 5 times table but with each term negative
3. Now you ask yourself: “what do I have to do to get from my -5 times table, back to my
sequence?”… well, be very careful because of the negatives, but you must… add 26 each time!
So, our rule for the nth term is…
5n  26
Important: Again, let’s use our rule to predict. How about the 6th term:
When n = 6
-5n + 26
-5 x 6 +26
=
… -4!
Well, the 6th term was the next one in our sequence, and I reckon if we had worked it out in our
head we would have got -4, so I think we are correct!
3. Writing out the Terms of a Sequence given the Rule
Now, so long as you understand what n means, you will be fine with this!
Just to re-cap, n is just the position of the term in the sequence.
So… if you want the 5th term, then n must be 5!
Example 1
Write out the first 5 terms of the sequence whose nth term rule is: 7n - 3
Okay, to get out 1st term, n must equal 1. So we have:
When n = 1
7n – 3
7x1–3
=
…4
so, 1st term is 4
=
… 11
so, 1st term is 11
Now to get out 2nd term, n must equal 2. So we have:
When n = 2
7n – 3
7x2–3
And if you keep this going, you end up with the first 5 terms: 4
11 18 25 32
Notice: the gap between each term is +7… which is what we would have expected from the 7n!
Example 2
Write out the first 5 terms of the sequence whose nth term rule is: n2 + 10
Looks hard, but same technique! To get out 1st term, n must equal 1. So we have:
When n = 1
n2 + 10
12 + 10
=
… 11
so, 1st term is 11
=
… 14
so, 1st term is 14
Now to get out 2nd term, n must equal 2. So we have:
When n = 2
n2 + 10
22 + 10
And if you keep this going, you end up with the first 5 terms: 1
14 19 26 35
4. Work out the nth term of Quadratic Sequences
Now, not all sequences are nice little linear ones.
If you have a look at the last example, you will see that the terms do not go up by the
same amount, and if you look at the nth term rule, you will see why… it’s quadratic!
Now, there is a really complicated method for finding the nth term of quadratics, but 9
times out of 10, a much simpler method works, so long as you know your square numbers!
1. Write out the square numbers (n2) underneath your sequence
2. Work out what you have to do to the square numbers to get back to your sequence
Example
Find the nth term of the following sequence:
-2
1
6
13
…
22
…
1. The terms are NOT going up by the same amount each time, so we need the square numbers…
n
1
2
3
4
5
…
…
Sequence:
-2
1
6
13
22
…
…
n2
1
4
9
16
25
…
…
2. What do you have to do to get from your square numbers back to your original sequence?…
well, I reckon you need to… subtract 3!
So, our nth term rule is:
n 3
2
Good luck with
your revision!