with the notion of a sequence.

列（数列）について学びます

And all a sequence is is an ordered list of numbers.

数列は 数字の順序リストです

So for example, I could have a finite sequence--

例えば 有限数列があります

that means I don't have an infinite number of numbers

つまり 無限数は含まれないものです

in it-- where, let's say, I start at 1 and I keep adding 3.

例えば １から始めて ３ずつ増やします

So 1 plus 3 is 4.

1＋3＝4

4 plus 3 is 7.

4＋3＝7

7 plus 3 is 10.

7＋3＝10

And let's say I only have these four terms right over here.

この4つだけが入っている数列を考えてみましょう

So this one we would call a finite sequence.

これを有限数列と呼びます

I could also have an infinite sequence.

無限数列もあります

So an example of an infinite sequence--

無限数列の例を挙げると

let's say we start at 3, and we keep adding 4.

3から始めて 4ずつ増やします

So we go to 3, to 7, to 11, 15.

よって 3、7、11、15となります

And you don't always have to add the same thing.

いつも同じ数字を足す必要はありません

We'll explore fancier sequences.

もっとシャレた数列も後で見ましょう

The sequences where you keep adding the same amount,

同じ数字を足し続ける数列を

we call these arithmetic sequences,

等差数列と呼びます

which we will also explore in more detail.

あとでもっと詳しく見ましょう

But to show that this is infinite,

この数列が無限だということを

to show that we keep this pattern going on and on and on,

このパターンが続くと示すために

I'll put three dots.

点を3つ書きます

This just means we're going to keep going on and on and on.

数列が続いていることを意味します

So we could call this an infinite sequence.

これを無限数列と呼びます

Now, there's a bunch of different notations

これとは違う、お洒落なやり方で

that seem fancy for denoting sequences.

数列を表す方法もあります

But this is all they refer to.

要するにこの数列の中にはこの数字が含まれています

But I want to make us comfortable with how

数列の表記法と 定義方法に

we can denote sequences and also how we can define them.

慣れていきましょう

We could say that this right over here

まず ここに

is the sequence a sub k for k is going from 1 to 4,

数列 a_k、kは1から4までが

is equal to this right over here.

以下と等しい、と書けます

So when we look at it this way, we

このように見ると

can look at each of these as the terms in the sequence.

この数字が数列内の項になります

And this right over here would be the first term.

これは第1項です

We would call that a sub 1.

a_1と呼びます

This right over here would be the second term.

これが第2項です

We'd call it a sub 2.

a_2と呼びます

I think you get the picture-- a sub 3.

大体わかりましたね、a_3

This right over here is a sub 4.

a_4です

So this just says, all of the a sub k's from k equals 1,

この表記法が言いたいことは、全てのa_kについて

from our first term, all the way to the fourth term.

K=1からK=4まで、つまり第1項から第4項まで、ということです

Now, I could also define it by not explicitly writing

数列を展開して書かない方法もあります

the sequence like this.

こんな風に全部書かなくても大丈夫

I could essentially do it defining our sequence

数列を関数として定義できます

as explicitly using kind of a function notation or something

定義を関数表記で書けます

close to function notation.

あるいは関数に近いもので。

So the same exact sequence, I could define it

全く同じ数列を 定義するには

as a sub k from k equals 1 to 4, with-- instead of explicitly

a_kについて、k=1 から4までで、

writing the numbers here, I could say a sub k

数字を書くのではなく

is equal to some function of k.

a_k = 「kの関数」と書きます

So let's see what happens.

どうなるでしょうか

When k is 1, we get 1.

k=1 のとき 1になります

When k is 2, we get 4.

k=2 のときは 4です

When k is 3, we get 7.

k=3のとき 7です

So let's see.

すると

When k is 3, we added 3 twice.

k=3 のとき 3を2回足しました

Let me make it clear.

分かりやすくすると

So this was a plus 3.

ここが+3で

This right over here was a plus 3.

ここも+3

This right over here is a plus 3.

隣も+3

So whatever k is, we started at 1.

よってkがいくつでも、1から始めました

And we added 3 one less than the k term times.

それから 3を (k−1) 回足しました

So we could say that this is going to be equal to 1

よって関数を書くと...

plus k minus 1 times 3, or maybe I

1 + 3 × (k−1)

should write 3 times k minus 1-- same thing.

1 + 3(k−1)

And you can verify that this works.

1 + 3(k-1)

If k is equal to 1, you're going to get 1 minus 1 is 0.

確かめてみましょう

And so a sub 1 is going to be 1.

式に当てはめたら確かめられます

If k is equal to 2, you're going to have 1 plus 3, which is 4.

k=1のとき 1-1=0

If k is equal to 3, you get 3 times 2 plus 1 is 7.

よって a_1=1 になります

So it works out.

k=2のとき 1+3=4 になります

So this is one way to explicitly define our sequence with kind

k=3のとき 3×2+1=7 になります

of this function notation.

正しい式になっていますね

I want to make it clear-- I have essentially

これが数列を関数表記のような

defined a function here.

形で定義する方法です

If I wanted a more traditional function notation,

つまり 私はここで

I could have written a of k, where

関数を定義したということです

k is the term that I care about. a

もっと伝統的な関数表記にしてもいいです

of k is equal to 1 plus 3 times k minus 1.

a(k)

This is essentially a function, where

kが今問題になっている項です

an allowable input, the domain, is

a(k)=1+3(k−1)

restricted to positive integers.

これは関数で

Now, how would I denote this business right over here?

代入可能な領域は

Well, I could say that this is equal to--

正の整数だけです

and people tend to use a.

こちらではどう表せるでしょうか

But I could use the notation b sub k or anything else.

これも等式になります

But I'll do a again-- a sub k.

a を使うことが多いです

And here, we're going from our first term--

b_kや 他も使えますが

so this is a sub 1, this is a sub 2--

a_kにしておきます a_k

all the way to infinity.

第1項は

Or we could define it-- if we wanted to define it explicitly

これがa_1 これがa_2

as a function-- we could write this sequence as a sub k, where

無限大まで続きます

k starts at the first term and goes to infinity,

陽関数として定義することもできます

with a sub k is equaling-- so we're starting at 3.

数列a_kとして

And we are adding 4 one less time.

kが第1項から始まり 無限大まで続きます

For the second term, we added 4 once.

a_k= 3 から始めて...

For the third term, we add 4 twice.

4を (k−1) 回ずつ足していきます

For the fourth term, we add 4 three times.

第2項に4を足して

So we're adding 4 one less than the term that we're at.

第3項に4を2回足します

So it's going to be plus 4 times k minus 1.

第4項には 4を3回足します

So this is another way of defining

よって 項の番号マイナス1回ずつ4を足していきます

this infinite sequence.

よって 3+4(k−1) と書けますね

Now, in both of these cases, I defined it

3+4(k−1)

as an explicit function.

これがもう1つの

So this right over here is explicit.

この無限数列の表記方法です

That's not an attractive color.

この両方の場合を陽関数として

Let me write this in.

定義しました

This is an explicit function.

こっちは陽関数です

And so you might say, well, what's

あまり良い色ではないですね

another way of defining these functions?

この色にしましょう

Well, we can also define it, especially something

これは陽関数です

like an arithmetic sequence, we can also define it recursively.

次に気になるのは

And I want to be clear-- not every sequence can be defined

この関数の他の表記方法ですね

as either an explicit function like this,

次のようにも定義できます

or as a recursive function.

等差数列のような関数は 帰納的に表すこともできます

But many can, including this, which

全ての数列が

is an arithmetic sequence, where we

このような陽関数で定義できるわけではないです

keep adding the same quantity over and over again.

帰納的関数でも同じです

So how would we do that?

しかし 多くはできます

Well, we could also-- another way of defining

例えばこれは等差数列で

this first sequence, we could say a sub k,

同じ量を足し続けています

starting at k equals 1 and going to 4 with.

では どうすればよいのでしょう

And when you define a sequence recursively,

他の方法は

you want to define what your first term is, with a sub 1

最初の数列はa_k

equaling 1.

k=1から始まり 4まで続きます

You can define every other term in terms of the term before it.

数列を帰納的に定義するには

And so then we could write a sub k

第1項をまず定義します

is equal to the previous term.

a_1=1

So this is a sub k minus 1.

他の項も その前の項を単位として定義します

So a given term is equal to the previous term.

よって、以下のように書けます

Let me make it clear-- this is the previous term, plus-- in

a_k=前の項

this case, we're adding 3 every time.

a_k=a_(k−1)

Now, how does this make sense?

今問題になっている項=前の項

Well, we're defining what a sub 1 is.

つまり 前の項 + ...

And if someone says, well, what happens when k equals 2?

この場合 3を毎項に足していきます

Well, they're saying, well, it's going to be a sub 2 minus 1.

a_(k-1)+3

So it's going to be a sub 1 plus 3.

さて どう成り立っているか見てみましょう

Well, we know a sub 1 is 1.

まず a_1を定義します

So it's going to be 1 plus 3, which is 4.

k＝2になると どうでしょう

Well, what about a sub 3?

a_(2-1) ですね

Well, it's going to be a sub 2 plus 3. a sub 2,

よって a_1+3 になります

we just calculated as 4.

a_1は1です

You add 3.

よって 1+3、つまり 4 になります

It's going to be 7.

a_3 はどうでしょうか

This is essentially what we mentally

a_2+3 になりますね a_2

did when I first wrote out the sequence, when I said, hey,

4ですね

I'm just going to start with 1.

3を足します

And I'm just going to add 3 for every successive term.

7になります

So how would we do this one?

これは 私達が最初に頭のなかでやったことと全く同じですね

Well, once again, we could write this as a sub k.

数列を書き始めたときに

Starting at k, the first term, going

1から始ようと言って...

to infinity with-- our first term, a sub 1,

その次に続く項に3を足し続けます

is going to be 3, now.

これはどうでしょう

And every successive term, a sub k,

もう一度 これをa_kとして書きましょう

is going to be the previous term, a sub k minus 1, plus 4.

kから始めて 第1項から無限大へ

And once again, you start at 3.

第1項 a_1=3

And then if you want the second term,

a_1=3

it's going to be the first term plus 4.

次に続く項は a_k

It's going to be 3 plus 4.

前の項 a_(k-1)+4

You get to 7.

３からもう1度始めます

And you keep adding 4.

第2項は

So both of these, this right over here

第1項 +4 になります

is a recursive definition.

3+4 です

We started with kind of a base case.

7になります

And then every term is defined in terms of the term

4を足し続けます

before it or in terms of the function itself,

これら2つの式は

but the function for a different term.

再帰的な定義です