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1+2+3+・・・+5000=?

少し前に数列に触れましたが、その超基礎にもう一度言及してみたいと思います。なぜなら、これから英語で行う、『1+2+3+・・・+5000=?』の計算方法の説明の中に用いられる言い回しや単語の中には、皆さんの今後の英語の勉強にお役に立つものが少なからずあるだろうと信じるからです。それではなるべく辞書を使わずに読みこなすように頑張ってください。

If you wrote out all the numbers from 1 to 5000

1~5000までの数字を書いたら、その下に逆順で
and then wrote them backwards underneath, you

同じ数字を書いてみて下さい。なんだか余計な
would have twice as many numbers as you

事をしたみたいですね。同じ数が2個ずつありますから
needed, but the problem is easier, here's why:

全部合わせると、求める和の2倍になります。でも

気にすることはありませんよ。なぜなら:

   1      2     3  .............. 4998 4999 5000
5000 4999 4998  ...............3      2      1
----------------------------------------
5001 5001 5001 ............. 5001 5001 5001
Notice that if we add the two lists, we get a list
that is the same number, 5001, repeating. In fact,
since each of the lists is 5000 numbers long, we
have, in the sums, a list of 5000 numbers that are
each 5001. さあ、5001が5000個もできてしまいました。
You have to admit that adding 5000 5001's is a lot
quicker than the other way since
5000 x 5001 = 25,005,000.

でも、これって必要な数の2倍分ですよね。
But wait, you say, that's too much. We were only
supposed to add one list and we added two. Okay,
then the answer must be half as much:
1 + 2 + 3 + ... + 5000 = 25,005,000/2 = 12,502,500
See, you can do the whole thing in your head.
You can use this method if you have any number
of consecutive numbers, whether they start with 1
or not. In fact, the numbers do not have to even be
consecutive. They just have to be in an arithmetic
sequence, that is, the difference between any two
adjacent numbers must always be the same (in your
example the difference was 1).

この調子でやると、この種の足し算はいくらでも

出来ますね。
Let's add up all the odd numbers from 1 to 25.
We do the same thing as before:
1 3 5 ......... 21 23 25
25 23 21 ......... 5 3 1
-----------------------------------

26 26 26 .......... 26 26 26
We have to know how many numbers there
are, and in this case there are 13 twenty-sixes,
so the total must be 13*26/2.
In general if you have an arithmetic sequence
of N numbers and you know the first and last one,
you can find the sum by:
Sum = N(First + Last)/2
Using this for your original set with
N = 5000, first = 1 and last = 5000,
we get the same result 5000(5001)/2.

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