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1b:Tf02,センター試験の集合は、実数の集合を扱うことが多いため、数直線上に図示するのが有効なことが多いです。
目盛の間隔を正確に図示する必要はなく、それぞれの端の大小と、黒丸白丸があっているかが重要です。(黒丸の場合はその点を含む、白丸の時はその点を含まないことを表します。不等号に=が入っているかどうかの違いとも言えます。)


例えば、
p: x>1
q:x≦2
のように与えられていた時、右向きの数直線上に左から1と2の点を書きます。

pについては、x>1(つまり「xは1より大きい」)であることから、先ほど書いた1の点に白丸を書き、そこから右上がりに少し直線を書き、そこから右向きに直線を伸ばします。新幹線のような形になります。この形は、1の点を含まないことを表すもので、白丸と同じ意味ですが、ぱっと見で分かるように両方使います。また、この線がpであることをどこかに書いておいてください。

qについては、x≦2(つまり｢xは2以下｣)であるので、2の点に黒丸を書き、そこから真下に少し直線を書き、左向きの直線を伸ばします。こちらは、電車のような形になります。この形は、2を含むことを表すもので、黒丸と同じ意味です。こちらの線にも、qであることを書いておいてください。

このように、範囲を一つ一つ図示していくと、次のようになります。
                      _______________ p
                    /           2
  ---------○-----●------->x
                   1           |
q --------------- 

これを見れば、「pかつq」や、「pまたはq」「p⇒q は真か偽か」はすぐに分かるはずです。たとえば｢pかつq｣なら、pとqが重なっているところなので、1<x≦2になります。｢pまたはq｣ならば、pとqの少なくともどちらかがある範囲なので、xは全ての実数になりますね。｢p⇒qは真か偽か｣については、pの中にqが含まれていないので、pならばqとはいえません。よって、偽となります。

上図の縦棒や斜め棒の長さを条件ごとに変えれば、一つの数直線にもっとたくさんの条件を書き込めます。そのようにして、一つの数直線に与えられた条件全てについて書いておくと、かなり簡単になると思います。

また、「(pかつq)または(rの否定)」といわれたときは、pとqとrとは別に、「pかつq」や｢rの否定｣についても書くと、分かりやすくなります。


加えて、たまに、条件式をそのまま使うと面倒くさいことがあります。そういう場合は、対偶を取るのが良いです。(そこまで多くはないし、絶対になければ解けないわけではないため、これ以後ついては忘れても大丈夫です)

「p⇒q」と、「(qの否定)⇒(pの否定)」(対偶)は同じ意味です。また、[(aかつb)の否定]と[(aの否定)または(bの否定)]は同じ意味です(ド・モルガンの法則)。これらをつかうことで、
・｢または｣を｢かつ｣に変換できる
・aやbの代わりにaの否定やbの否定を使える
という利点があります。このような利点が使えそう！と思ったら使ってみてください(とりあえずわかんなかったら対偶とってみる、っていうのも一つの手ではあります)。

※(rの否定)などは、本来はrの上に横棒を書いて表します

至らないところもあったかもしれませんが、貴方の合格を願っています。それでは。1d:Tc1f,　こういった問題独自の定義は、だいたい文字を含んでいることが多いです。例えば、
・「nを正の整数とし、3^nを10で割った余りをanとする。」（東京大2016文系）
・「正の整数nの各位の数の和をS(n)で表す。」（一橋大2018）
・「nを2以上の整数とする。金貨と銀貨を含むn枚の硬貨を同時に投げ、裏が出た金貨は取り去り、取り去った金貨と同じ枚数の銀貨を加えるという試行の繰り返しを考える。初めはn枚すべてが金貨であり、n枚すべてが銀貨になった後も試行を繰り返す。k回目の試行の直後に、n枚の硬貨の中に金貨がj枚だけ残る確率をPk(j)（0≦j≦n）で表す。」（東北大2019文系）
のように。あなたが挙げて下さった例でもそうですね。
　ご存知のように、数学で文字が使われるのはそこに入る値が不特定であるときなので、逆にいえば、自分で具体的な値を代入して実験してみれば良いわけです。k-連続和でいえば、m=1、k=2とすると、3＝1+2という等式になり、3は2-連続和であることになります（相談文のk+1はおそらくkー1の間違いですね。でなければ、nはk+2個の連続する自然数の和になってしまうので）。ちゃんと、n(3)がk(2)個の連続する自然数(1→2)の和であるという定義に則ってますね。2019年文系の確率も、例えばk＝1を代入してみると、P1(j)は「n枚の金貨を同時に投げ、そのうちj枚が表で他が裏になる確率」のことを言っているのだとわかります（ちなみにこれは小問⑴）。反復試行の確率を考えればすぐ解けますね。すると、次はk＝2、その次はk＝3、と実験数をどんどん増やしていけば、Pk(j)の内容もいずれわかるはずです。試行の手順上、残るj枚は必ず全ての試行において表でなければならず、他方それ以外の金貨はすべて、k回のうちのどこかで裏が出ればいい（全て表で残る場合の余事象）わけですから、「n枚の金貨のうち、k回の試行の直後に残るべきj枚はk回とも全て表が出て、それ以外のn−j枚はk回の試行で少なくとも一回裏が出る確率」とわかります。ここまで日本語として簡略化できれば、Pk(j)（特に、k≧2）の値もそこまで苦戦せずに出せそうですね（ちなみにこれは小問⑵）。
　このように、なるべく簡単な値から代入して実験を繰り返すことで、独自の定義が何を言っているのかは帰納的に理解できることが多いです。文字が多かったり、分かりにくい表現だったりして、複雑で難しく感じる定義が出てきたら、まずは実験してみることを心がけると良いと思います。文系の問題ですが、もしまだ解いてない場合はネタバレになってしまい申し訳ございません。1e:Tf9c,数学を根本的に理解する。
という勉強方法は、言葉で説明すると少し難しいので、ほんの少しだけここでやっていみたいと思います。


例えば、弧度法の中で「ラジアン」というのが出てくると思います。これは、「2π = 360°」を基準に考えよう。という風に習ったと思います。このラジアンを使って、扇形の弧の長さを求める公式で、「L = rθ」というのがあります。
皆さんの中に、この式を覚えているだけになっていて、意味を理解していない方はおられるでしょうか？

これは、小学校の時に習った、「円周の長さは2πr」というものを使っています。

どういうことかと言うと、「円を4分割した形である扇形のこの長さを求めよ。」という問題があった時、

小学校で習った式を使うと、求めるのは円周を4等分した長さなので、 ¼ × 2πr = ½πr

ラジアンを使って解くと、中心角 90° は、ラジアンでは ½π なので、L = r × ½π = ½πr 

よって、答えはどちらの式を使っても、½πr になりました。

中学の知識では、L = 2r × π × 角度 / 360°
高校数学では、L = rθ
どちらの公式でも求められますが、公式で見ると、弧度法を使った方が分かりやすいですよね。



という感じです。

公式をただ覚えるだけでなく、意味を理解しながら使えるようになる。ということが、根本的に理解するということになります。

先程の例で言うと、ラジアンというものはどういう意味を持つのか。ラジアンを使えるようになると、計算がどう変わるのか。というのを理解しておく必要があります。



これは、ほかの公式でも当てはまります。

例えば、加法定理の公式：
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
これを使って2倍角の公式を作ります。
sin2a = sin(a+a) = sin(a)cos(a) + cos(a)sin(a)
          = 2sin(a)cos(a)


例えば、等差数列の和の公式：
S = ½n(a + l) (a：初項、l：末項、n：項数)
これに、末項：l = a + (n - 1)d (d：交差) を代入すると、
S = ½n(2a + (n - 1)d)
これが教科書に乗っている和の公式の2つになります。


こんなん知ってるよ。という方もいるかもしれません。ただ、これが数学を根本的に理解するということになります。

もう少し難しい話に行くと、
・解の公式ってなんであの形なの？
・平方完成ってなんでするの？
・円の方程式の意味は？
・微分と積分の関係は？
・ベクトルって何？
などなど……
キリがないので、この辺りにしておきますが、




要するに、公式の意味を理解することで、数学を本質的に理解しよう。という訳です。

しかも、これらは全てほとんどの教科書に載っています。理解しようと思うと、教科書を読めば大体のことが分かります。


数学を根本的に理解すると、問題を解くときに答え方がパッと思いつきやすくなると思います。さらに、公式の丸暗記では、時間が経つと忘れてしまうかもしれませんが、理論的に覚えていると、脳の構造的にも忘れにくくなるということもあります。なので、この勉強方法をオススメする方はたくさんいますし、私もこのやり方で勉強しました。

ただ、人によっては向き不向きがありますので、これを絶対に使った方がいいとは私は言えません。

実際に、私もこれで苦手だった数学が、だんだんと解けるようになったので、興味があれば、是非やってみてください。

長文失礼しました。是非参考になればと思います。1f:Tbc4,ご質問にお答えさせていただきます！東京大学理科一類現役合格の者です。

進研模試の数学の偏差値が64ほどということは、そこまで基礎がなっていないと言うことでもないように感じます。

現在高２ということはあと数ヶ月ほどで高３ですよね。京大志望ということであれば時間がとにかくないので、はっきり言って今の時期からの基礎問題精講は時間の無駄のように感じます。なおさら貴方のようにある程度できているようであればなおさらです。

もしそれでも問題が難しくて中々解き進められないと言う場合は、その分野の青チャートの例題をササッと確認して基礎を見直すと言うのが効率の良い勉強法だと思います。

また、とにかく解いていて楽しいと言うことであれば必ず成長できると思いますよ！苦でなければ人はある程度のことは続けられます。

ただ注意点として数学の解答例を見るときは式の操作の意味（目的）を常に意識してよむようにしてください。ここに大きな勉強の質の差が生まれると私は思っています。
簡単なたとえですが、放物線の二次式を見たら大抵の人は平方完成をまず行うでしょう。
ではそれはなぜでしょうか？
私たちは放物線を始めに学習したときにy=x^2からまず習い、次にy=x^2+cのy方向への平行移動を、そしてy=(x+b)^2のx方向への平行移動を、最後にy=ax^2の放物線の開き具合について習ったかと思います。これらをすべて組み合わせたのがy=a(x+b)^2+cという式になり放物線に関する諸情報が得られる訳です。

こんな風に解答にある式変形は「何の情報をどんな手段で導こうとしているのか」を常に意識し理解し自分のものに落とし込みましょう。ぱっと分からなかった場合は自分で書き込んでおくのもいいかもしれません。

また、解いた問題には何か記しやコメントを書いておくといいと思います。私の場合は、☆key問題、○普通に解けた、△少し迷ったけどなんとか解けた、×解けなかった、そのほかにも「良問！」「なるほど！」「分かるか～！」（コメントは割と自由）など書いていました。
そうすると復習をするときに見返しやすいですし、思い出しやすいように感じています！

とりあえず標準問題精講レベルは春休みの間に修了することを目標にして、入試問題の王道的な解き方を習得しましょう。そうすれば高３から少し上の問題集や志望校の過去問演習にスムーズに取り組むことができるはずです。

他に何か質問があれば何なりとしてください。応援しています！

2:["$","main",null,{"className":"px-4 pt-4 pb-4","children":["$","div",null,{"className":"max-w-3xl mx-auto w-full","children":[["$","div",null,{"className":"mb-8","children":["$","$L7",null,{"href":"https://unilink-app.onelink.me/isbO/h6xeh63x?advice=AlLHe4EBTqPwDZPuGZFB","target":"_blank","children":["$","$L8",null,{"src":"/images/web_to_app_banner.jpg","width":3660,"height":1500,"sizes":"100vw","style":{"width":"100%","height":"auto"},"alt":"UniLink WebToAppバナー画像","className":"mb-4 rounded"}]}]}],["$","h1",null,{"className":"text-xl font-semibold mb-2","children":"数Ⅱ　図形と方程式　線分の内分点の軌跡"}],["$","div",null,{"className":"flex justify-between mb-4","children":[["$","div",null,{"className":"text-left text-xs text-caption","children":["クリップ(",0,") コメント(",4,")"]}],["$","div",null,{"className":"text-right text-xs text-caption","children":"6/19 20:44"}]]}],["$","div",null,{"className":"coach-mark 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"隣接3項間漸化式"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"こんにちは、名古屋大学医学部医学科のメイメイといいます。\n(an-an-1)=bnとするとb1は求められないですね。\n\n(an+1)-(an)=2［(an)-(an-1)］\nが出てきているはずですが、\n\nn-1の項があり基本的にn≧2で考えています。\nこれをn≧1に直してみると\n(an+2)-(an+1)=2［(an+1)-(an)］\nとなります。\n単純にnの部分を1ずつずらしただけです。\n\nこの状態で(an+1)-(an)=bn\nと置いてみましょう。\n\nb1が求められるはずです。(ちなみにb2は必要ないです。)\n\nつまり(bn+1)=2(bn)、b1=(a2)-(a1)=8の等比数列に帰着しますね。\n\nこれを解くと、bn=8・2^n-1=2^n+2となります。(2^n-1は2のn-1乗という意味です。)\n\nすなわち、(an+1)-(an)=2^n+2\n\n両辺を2^n+1で割ると\n\n<(an+1)/2^n+1>-(1/2)<(an)/2^n>=2\n\nとなります。\n\n(an)/2^nをcnとすると、(cn+1)=(1/2)(cn)+2\n\nこれを変形して、(cn+1)-4=(1/2)<(cn)-4>\n\nつまり(cn)-4=(-7/2)・(1/2)^n-1=(-7)・(1/2)^n\n\nよってcn=4-7・(1/2)^n\n\nこの両辺に2^nをかけてan=4・2^n-7 (n≧1)\n\nとなります。\n分かりにくくてすいません！\n\n"}],["$","div",null,{"className":"flex 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"暗記数学について"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"こんにちは。受験数学は暗記とよく言われますよね。私の考える暗記数学の定義とやり方について説明します。\n\n私は、暗記数学とは｢ある特徴を持った数学の問題に対処出来る解法パターンやテクニック、公式などを答えを見て覚え、その知識を使って解く数学｣だと考えます。\n\n例を用いて説明しましょう。例えば数学Aの数学と人間の活動(旧:整数の性質)で、｢x+y+z=xyzを満たす自然数x, y, z(ただしx<y<z)の組を全てを求めよ｣という問題があります。\n\nこれは不等式評価の典型問題ですが、恥ずかしながら私は初見で解法を思いつくことはできませんでした。ちなみにこの問題はx<y<zを用いてxyz=x+y+z<z+z+z=3zから、xy<3を導いて解きます。答えは(1, 2, 3)です。\n\nこの問題の暗記すべきポイントは｢このような問題に対しては、和の項に最も大きい文字を代入して上から評価する(xy<3とすることで絞る)こと｣でしょうか。初見の私にはこのような不等式を立てて評価するという発想がありませんでした。しかし、解答を読んだら次からこういった問題はこう解けばいいと分かりました。\n\nこのように、初見だと思いつくのは難しいけれども解答を見たら理解出来る問題が高校数学には大量に出てきます。そりゃあ青チャートがあんなに分厚くなるわけです。典型問題に対して1問1問熟考するよりも、解答を見てパターンを暗記し、次から問題を解けるようになる方が(チャートなら)効率がいいのです。\n\nやり方も多様ですが、私の場合①問題を読んで解法を考える→②解法が思いつかなかったら印をつけて解答を見る→③パターンを理解する→④類題を解く→⑤時間をあけて①へ戻る、の繰り返しをしていました。段々慣れてくると適当になってしまったため、2周目の①では手を動かして理解していることを確かめていました。\n\n最後になりますが、暗記数学は高校1, 2年生で終わらせて置くといいです。高校3年生は過去問の演習や模試などのアウトプットで忙しくなりますので、それまでに典型問題の解法をインプットしておくと大きなアドバンテージになります。進研模試を受験なさっていると思いますが、進研模試の数学は各大問の(2)まではだいたい解法暗記で解けます。(3)も応用せずとも太刀打ちできるのもあります。\n\nこの回答が学習の助けとかれば幸いです。勉強応援しています！！\n"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 24","className":"text-subPrimary mr-1","children":["$undefined",[["$","path","0",{"fill":"none","d":"M0 0h24v24H0V0z","children":[]}],["$","path","1",{"d":"M12 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"証明や導出がすごい気になってしまう"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"三味線さん、はじめまして。\n\nお気持ちはすごく分かります。\nたしかに解答の細かいところに疑問を持ったり、その都度公式を導出していると参考書の進むペースは遅くなってしまいますが、その分、質は高くなると思うので全然良いことだと思いますし、むしろそうするべきだと思います。\n\nよく言われる「数学は理解」という言葉は、なぜその公式を使ったのか、なぜその解法で解くのか、なぜその変換を行うのか、もっと細かいことで言うと、なぜその順に解答を記述するのかといったことを理解することです。\n\n「数学は暗記」という言葉もたまに聞きますが、これは単純に英単語みたいに暗記すると言うことではなくて、どうしてこの解法を使うのかを理解した上でどうゆう問題が出たらどの解法を使うのかを暗記すると言うことです。\n仮に理解の過程を飛ばして暗記だけすると、少し問題の形が変わっただけで解法が思い浮かばないということになってしまいます。\n\nそして理解を深めるためには、三味線さんのように細かいところにも疑問を持って問題を解くのが一番の近道です。公式は導出ができる方が理解度ははるかに上がりますし、たまにある公式の導出に基づいた問題なんかも出題されることもあります。\nまた質問文中のことで触れると、なぜ置換積分はこうゆう形でするのか、一次独立とは何か、解答に使われている言葉の意図、こういったことに疑問をもって考えるのはとても良いことだと思います。確認しても忘れてしまうのは人間なので仕方ないことで、確認してその時に理解したことをノートなんかに纏めておきましょう。次に同じような疑問が出た時にノートを見返すことで少しずつ定着して力になっていくはずです。\n私の場合だと2.3回では定着せず、5回とか10回その都度見返すことで定着し始めた感じだったので、忘れているから力になっていないと焦らずに、自分のペースで頑張ってください！\n\n応援しています☺️\n"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 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