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漸化式の解き方と数学的帰納法の活用

数列の中でも特に応用力が求められる「漸化式と数学的帰納法」について学んでいこう。漸化式は隣り合う項の関係から一般項を求める手法で、数学的帰納法は自然数に関する命題を証明する強力な道具だ。この2つは共通テストでもよく出題されるから、基本パターンをしっかりマスターしておこう。

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# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

漸化式と数学的帰納法の基礎

君が数列で最も苦戦するのがこの分野かもしれないが、実はパターンを覚えれば確実に解けるようになる。漸化式は数列の隣り合う項の関係式で、初項と一緒に与えられると数列全体が決まる仕組みだ。

一般項を求めることが最終目標になる。例えば a1=2a_1 = 2an+1=2an+1a_{n+1} = 2a_n + 1 という漸化式があれば、これから ana_n をnの式で表すことを目指す。

一方、数学的帰納法は自然数nに関する命題を証明する方法だ。ドミノ倒しに例えられることが多く、最初のドミノ(n=1)を倒し、あるドミノが倒れれば次も倒れることを示せば、すべてのドミノが倒れることを証明できる。

覚えておこう: 共通テストでは誘導に従って解き進める形式が多いから、基本パターンの習得が何より重要だ。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

漸化式の4つの基本パターン

漸化式の問題は型を見抜くことが勝負の分かれ目だ。まずは形を判別してから適切な解法を選ぼう。

等差数列型an+1=an+da_{n+1} = a_n + d)と等比数列型an+1=rana_{n+1} = ra_n)は基本中の基本。それぞれ公差d、公比rを使って一般項がすぐに求まる。

階差数列型an+1=an+f(n)a_{n+1} = a_n + f(n))では、階差数列の考え方を使って an=a1+k=1n1f(k)a_n = a_1 + \sum_{k=1}^{n-1} f(k) で求める。ただし、この公式は n2n \geq 2 で成り立つから、n=1n=1 のときの確認を忘れずに。

最も重要なのが an+1=pan+qa_{n+1} = pa_n + qだ。特性方程式 α=pα+qα = pα + q を解いて、an+1α=p(anα)a_{n+1} - α = p(a_n - α) の形に変形する。この変形により等比数列の問題に帰着できる。

ポイント: 特性方程式を使う理由は、等比数列の形 bn+1=pbnb_{n+1} = pb_n を作るため。bn=anαb_n = a_n - α とおくことで解決できる。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

数学的帰納法の証明手順

証明問題は型通りに書くことで確実に点数が取れる。手順を守って、採点者に分かりやすく論理を展開しよう。

[I] n=1での成立: まず n=1n=1 を命題に代入して、左辺と右辺を計算する。両辺が等しくなることを確認して「n=1n=1 のとき命題は成り立つ」と結論づける。

[II] n=k⇒n=k+1の証明: 「n=kn=k のとき命題が成り立つ」と仮定し、この仮定を使って n=k+1n=k+1 のときも成り立つことを示す。仮定の式を明確に書いて、どこで使ったかを「仮定より」と明記することが重要だ。

例えば $1+2+...+n = \frac{nn+1n+1}{2}を証明する場合、 を証明する場合、n=k+1のときの左辺で のときの左辺で 1+2+...+kの部分を仮定の式 の部分を仮定の式 \frac{kk+1k+1}{2}$ に置き換えて計算を進める。

結論: 最後に「[I], [II]より、すべての自然数nについて命題は成り立つ」と締めくくる。この定型文を忘れると減点される可能性がある。

注意: 「仮定より」という一言と、仮定をどこで使ったかを明確にすることで、論理の流れが採点者に伝わりやすくなる。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

実際の問題で理解を深めよう

具体例を通して解法の流れを確認してみよう。a1=3a_1 = 3an+1=2an1a_{n+1} = 2a_n - 1 の一般項を求める問題を考える。

これは an+1=pan+qa_{n+1} = pa_n + q 型(p=2,q=1p=2, q=-1)だから、特性方程式 α=2α1α = 2α - 1 を解く。α=1α = 1 が得られる。

漸化式を変形して an+11=2(an1)a_{n+1} - 1 = 2(a_n - 1) とする。bn=an1b_n = a_n - 1 とおくと、bn+1=2bnb_{n+1} = 2b_n という等比数列になる。

b1=a11=31=2b_1 = a_1 - 1 = 3 - 1 = 2 だから、bn=22n1=2nb_n = 2 \cdot 2^{n-1} = 2^n。よって an=bn+1=2n+1a_n = b_n + 1 = 2^n + 1 が答えだ。

数学的帰納法の証明では、n=1n=1 での確認、仮定の明記、n=k+1n=k+1 での計算、結論という4つのステップを必ず踏む。計算ミスを避けるために、各ステップを丁寧に進めることが大切だ。

試験のコツ: 階差数列の公式を使った後は、必ず n=1n=1 のときも一般項が成り立つかをチェックしよう。これを忘れると大きく減点される。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

試験対策の最終チェック

試験で確実に得点するために、以下のポイントを押さえておこう。

漸化式のパターン認識: 問題を見たら冷静にどの型か判断する。等差、等比、階差、特性方程式の4パターンを瞬時に見分けられるよう練習を重ねよう。特性方程式では an+1α=p(anα)a_{n+1} - α = p(a_n - α) の変形を確実にマスターすることが重要だ。

数学的帰納法の記述: 証明の型を守ることが何より大切。「n=kn=k で仮定し、n=k+1n=k+1 で示す」という構造を明確に書き、仮定をどこで使ったかを必ず明記する。

計算の正確性: 特性方程式の変形や帰納法の n=k+1n=k+1 での計算は複雑になりがち。焦らず丁寧に展開・因数分解を行おう。不等式の証明では A>BA > B を示すために AB>0A - B > 0 を示すのが基本戦略だ。

この分野は練習量がものを言う。様々なパターンの問題に触れて解法の引き出しを増やせば、必ず得点源にできる分野だ。

合格への近道: 基本パターンを完璧にしてから応用問題に取り組もう。焦らず着実にステップアップすることが成功の秘訣だ。



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Paul T

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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

 

共通試験

15

Updated Apr 28, 2026

5 pages

漸化式の解き方と数学的帰納法の活用

数列の中でも特に応用力が求められる「漸化式と数学的帰納法」について学んでいこう。漸化式は隣り合う項の関係から一般項を求める手法で、数学的帰納法は自然数に関する命題を証明する強力な道具だ。この2つは共通テストでもよく出題されるから、基本パターンをしっかりマスターしておこう。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

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漸化式と数学的帰納法の基礎

君が数列で最も苦戦するのがこの分野かもしれないが、実はパターンを覚えれば確実に解けるようになる。漸化式は数列の隣り合う項の関係式で、初項と一緒に与えられると数列全体が決まる仕組みだ。

一般項を求めることが最終目標になる。例えば a1=2a_1 = 2an+1=2an+1a_{n+1} = 2a_n + 1 という漸化式があれば、これから ana_n をnの式で表すことを目指す。

一方、数学的帰納法は自然数nに関する命題を証明する方法だ。ドミノ倒しに例えられることが多く、最初のドミノ(n=1)を倒し、あるドミノが倒れれば次も倒れることを示せば、すべてのドミノが倒れることを証明できる。

覚えておこう: 共通テストでは誘導に従って解き進める形式が多いから、基本パターンの習得が何より重要だ。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

Sign up to see the contentIt's free!

Access to all documents

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Join milions of students

漸化式の4つの基本パターン

漸化式の問題は型を見抜くことが勝負の分かれ目だ。まずは形を判別してから適切な解法を選ぼう。

等差数列型an+1=an+da_{n+1} = a_n + d)と等比数列型an+1=rana_{n+1} = ra_n)は基本中の基本。それぞれ公差d、公比rを使って一般項がすぐに求まる。

階差数列型an+1=an+f(n)a_{n+1} = a_n + f(n))では、階差数列の考え方を使って an=a1+k=1n1f(k)a_n = a_1 + \sum_{k=1}^{n-1} f(k) で求める。ただし、この公式は n2n \geq 2 で成り立つから、n=1n=1 のときの確認を忘れずに。

最も重要なのが an+1=pan+qa_{n+1} = pa_n + qだ。特性方程式 α=pα+qα = pα + q を解いて、an+1α=p(anα)a_{n+1} - α = p(a_n - α) の形に変形する。この変形により等比数列の問題に帰着できる。

ポイント: 特性方程式を使う理由は、等比数列の形 bn+1=pbnb_{n+1} = pb_n を作るため。bn=anαb_n = a_n - α とおくことで解決できる。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

数学的帰納法の証明手順

証明問題は型通りに書くことで確実に点数が取れる。手順を守って、採点者に分かりやすく論理を展開しよう。

[I] n=1での成立: まず n=1n=1 を命題に代入して、左辺と右辺を計算する。両辺が等しくなることを確認して「n=1n=1 のとき命題は成り立つ」と結論づける。

[II] n=k⇒n=k+1の証明: 「n=kn=k のとき命題が成り立つ」と仮定し、この仮定を使って n=k+1n=k+1 のときも成り立つことを示す。仮定の式を明確に書いて、どこで使ったかを「仮定より」と明記することが重要だ。

例えば $1+2+...+n = \frac{nn+1n+1}{2}を証明する場合、 を証明する場合、n=k+1のときの左辺で のときの左辺で 1+2+...+kの部分を仮定の式 の部分を仮定の式 \frac{kk+1k+1}{2}$ に置き換えて計算を進める。

結論: 最後に「[I], [II]より、すべての自然数nについて命題は成り立つ」と締めくくる。この定型文を忘れると減点される可能性がある。

注意: 「仮定より」という一言と、仮定をどこで使ったかを明確にすることで、論理の流れが採点者に伝わりやすくなる。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

実際の問題で理解を深めよう

具体例を通して解法の流れを確認してみよう。a1=3a_1 = 3an+1=2an1a_{n+1} = 2a_n - 1 の一般項を求める問題を考える。

これは an+1=pan+qa_{n+1} = pa_n + q 型(p=2,q=1p=2, q=-1)だから、特性方程式 α=2α1α = 2α - 1 を解く。α=1α = 1 が得られる。

漸化式を変形して an+11=2(an1)a_{n+1} - 1 = 2(a_n - 1) とする。bn=an1b_n = a_n - 1 とおくと、bn+1=2bnb_{n+1} = 2b_n という等比数列になる。

b1=a11=31=2b_1 = a_1 - 1 = 3 - 1 = 2 だから、bn=22n1=2nb_n = 2 \cdot 2^{n-1} = 2^n。よって an=bn+1=2n+1a_n = b_n + 1 = 2^n + 1 が答えだ。

数学的帰納法の証明では、n=1n=1 での確認、仮定の明記、n=k+1n=k+1 での計算、結論という4つのステップを必ず踏む。計算ミスを避けるために、各ステップを丁寧に進めることが大切だ。

試験のコツ: 階差数列の公式を使った後は、必ず n=1n=1 のときも一般項が成り立つかをチェックしよう。これを忘れると大きく減点される。

# 漸化式と数学的帰納法

## 漸化式と数学的帰納法の概要

数列の単元で一番応用力が試される分野。漸化式は隣り合う項の関係から一般項
を求める問題で、数学的帰納法は自然数nに関する命題を証明するための強力
な手法。この二つはセットで出題されることも多いから、しっかり理解してお

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試験対策の最終チェック

試験で確実に得点するために、以下のポイントを押さえておこう。

漸化式のパターン認識: 問題を見たら冷静にどの型か判断する。等差、等比、階差、特性方程式の4パターンを瞬時に見分けられるよう練習を重ねよう。特性方程式では an+1α=p(anα)a_{n+1} - α = p(a_n - α) の変形を確実にマスターすることが重要だ。

数学的帰納法の記述: 証明の型を守ることが何より大切。「n=kn=k で仮定し、n=k+1n=k+1 で示す」という構造を明確に書き、仮定をどこで使ったかを必ず明記する。

計算の正確性: 特性方程式の変形や帰納法の n=k+1n=k+1 での計算は複雑になりがち。焦らず丁寧に展開・因数分解を行おう。不等式の証明では A>BA > B を示すために AB>0A - B > 0 を示すのが基本戦略だ。

この分野は練習量がものを言う。様々なパターンの問題に触れて解法の引き出しを増やせば、必ず得点源にできる分野だ。

合格への近道: 基本パターンを完璧にしてから応用問題に取り組もう。焦らず着実にステップアップすることが成功の秘訣だ。

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