What makes a sequence more than a bunch of numbers is the order the numbers feature.
Think of it as a cloak room of a bowling club, school play, theater, nightclub.
There is a ticket with a number for each item, well,
the analogy breaks down as the items are all different.
Two people could have the same coat or bag, the point is
the items in the sequence are labeled and so, there is an order.
In a different day, we have a different sequence of things.
When we deal with sequences of numbers, we care about the variation of numbers
as the labels, ticket numbers, increase.
So a sequence is an ordered list of numbers,
here's some examples of sequences.
Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday, ending there.
1, -1, 1, -1, 1, -1,
and so on, 3, 3, 3, 3, 3 and
carry on like that, 1, 11,
111, 1,111 and so on, or 5,
4, 3, 2, 1, 0, ending there.
As a computer scientist, one of the basic uses of sequences will be
tapping into their patterns in order to be able to use them computationally.
As I mentioned in the introduction, there are many other applications, but for
this video, let's be pattern hunters.
So take a moment to go through the sequences presented.
Make some notes on the patterns found in your own way of
describing each of the sequences in words.
Let's gather our thoughts on this,
the first is the sequence of week days starting on Monday.
Note that is a finite sequence, he only has seven terms.
The second sequence is infinite as implied by the dot, dot,
dot, which says continuing in the same pattern.
So the pattern is alternating, alternating between 1 and -1.
Or you could say, beginning in 1, subtract 2, then add 2, and so on.
Another way of saying it would be starting 1 and
alternates between subtracting 2 and adding 2.
There is more than one way of describing it,
the third sequence is a constant sequence of threes and is also infinite.
The fourth could be described as begin with,
then write another 1 next to it and carry on like this.
Or we could say instead on each position write that many one's.
So one 1, 1 in the first position, two one's in the second position,
three one's in the third position and so on.
Or you may be thinking of place value, add up consecutive piles of ten.
So starting 1 and add 10 for the next term, then at 100 and
so on, or you could even be thinking of binary.
Anyway, I let you think of how to express this pattern,
the fifth is a finite sequence, a count down from 5.
Or starting 5 and subtract 1 to get next, stop at 0,
let's have another look at sequences.
I want you to write down the patterns you find, so
that you can describe them to another person.
Here are the sequences, we can discuss them after the pause.
Now, the first is a sequence of a consecutive square numbers
starting from the square of one.
The second is a sequence of consecutive cube numbers starting from the cube of 1.
The third is a sequence of fraction, of ratios of consecutive numbers.
The first term is 1 over 2, the second term,
is the second number over the third, so 2 over 3, and so on.
So the nth term is n divided by n + 1.
We start needing formal mathematical language to define
the sequences precisely.
Now, the fourth is 0 point as many nine's are the order of the term.
Does that sound complicated?
The fifth could be the sequence of prime numbers, there is no formula for this.
The sixth sequence, seems to be a power of two and the number that comes next.
Then the next power of two and the number, one over it, and so on, and so on.
You probably with me, we need a better way of formalizing these patterns.
Join me in the next video, after the quiz.