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数学的帰納法の問題の解法 | 夢を叶える塾
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数学的帰納法の問題の解法

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数学的帰納法を使って証明する問題です。

基本問題

\displaystyle 1+2+...+n=\frac{n(n+1)}{2}を数学的帰納法を使って証明しなさい。

解き方

数学的帰納法とは、

  1. n=1のときに式が成り立つことを示す
  2. n=kのときに式が成り立つと仮定したとき、n=k+1のときに式が成り立つことを示す
  3. 1,2より全ての自然数nで式が成り立つ

というのが基本パターンです。
「n=1のときに成り立っていて、n=1が成り立つんだからn=2も成り立つよね。」
「n=2のときに成り立っているんだから、n=3も同じく成り立ち、n=4もn=5もずっと成り立つよね」
という考え方です。
中には、

  1. n=1,n=2のときに式が成り立つことを示す
  2. n=k,n=k+1のときに式が成り立つと仮定したとき、n=k+2のときに式が成り立つことを示す
  3. 1,2より全ての自然数nで式が成り立つ

というパターンもあります。
「n=1,n=2のときに成り立っていて、n=1,n=2で成り立つんだからn=3も成り立つよね。」
「n=2,n=3のときに成り立っているんだから、n=4も同じく成り立ち、n=5もn=6もずっと成り立つよね」
連鎖的に次々と成り立つことが言えるような状況を作り出します。

証明の方法です。
①n=1のときに成り立つのかどうかは、与えられた命題をn=1でどうなるか考えます。
n=1でその命題が真であることを言いましょう。
②次にn=kのとき成り立つとすると、何が言えるのかです。
与えられた命題をn=kでどうなるか考えます。
その命題が真であるという仮定をします。
つまりn=kのケースは成り立っているものとして考えます。
③そしてn=k+1のときに成り立つことを言います。
これはn=k+1のときの命題をn=kの情報を使って証明します。
④最後に数学的帰納法により成り立つという事を言います。

具体的な問題を使って証明方法を理解しましょう。

解説

\displaystyle 1+2+...+n=\frac{n(n+1)}{2}を数学的帰納法を使って証明しなさい。

まずn=1のときに成り立つことを証明します。
次にn=kのときに式が成り立つと仮定したとき、n=k+1のときに式が成り立つことを証明します。

(1)n=1のときに成り立つことを示す
\displaystyle 1+2+...+n=\frac{n(n+1)}{2}がn=1のときどうなるか?
左辺は、1からnまでの和です。
1から1までの和になりますので
左辺=1
右辺は式にn=1を代入しましょう。
右辺=\displaystyle \frac{1(1+1)}{2}=\frac{2}{2}=1
ですね。
左辺=右辺です。
よって成り立ちます。

(2)n=kのときに式が成り立つと仮定したとき、n=k+1のときに式が成り立つことを示す
\displaystyle 1+2+...+n=\frac{n(n+1)}{2}がn=kのときどうなるか?
そのままkで置き換えましょう。
\displaystyle 1+2+...+k=\frac{k(k+1)}{2}です。
kまでの和が\displaystyle \frac{k(k+1)}{2}で表せるという事は使っていいわけです。

n=k+1のときを考えてみましょう。
左辺\displaystyle =1+2+...+k+(k+1)
ですね。
右辺\displaystyle =\frac{(k+1)((k+1)+1)}{2}=\frac{(k+1)(k+2)}{2}
ですね。
左辺を変形して、この右辺の形になればいいわけです。
※右辺を変形して左辺でも良いですが

左辺\displaystyle =1+2+...+k+(k+1)を変形していきます。
まず\displaystyle 1+2+...+k=\frac{k(k+1)}{2}が成り立っています。
これを使って、
左辺\displaystyle =(1+2+...+k)+(k+1)=\frac{k(k+1)}{2}+(k+1)
式を変形していきます。
\displaystyle =\frac{k(k+1)}{2}+(k+1)
\displaystyle =\left(\frac{k}{2}+1\right)(k+1)
\displaystyle =\frac{k+2}{2}(k+1)
\displaystyle =\frac{(k+1)(k+2)}{2}
=右辺
よってn=kのときに式が成り立つと仮定したとき、n=k+1でも成り立ちます。

(1)(2)より数学的帰納法を用いてすべての自然数nについて与式は成り立つ。
で締めくくりましょう。

終わりに

定期試験などで出題される「数学的帰納法を使って証明しなさい」の問題はサービス問題です。
問題に答えが書いてあるんです。
証明すべきものは、数学的帰納法で証明できるものです。
そしてその証明方法の数学的帰納法を使う事まで書いてあります。
問題が答えを教えてくれている問題です。
数学的帰納法が試験範囲に入っていたらラッキーですね。

関連

数学的帰納法で倍数を証明する問題の解法

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