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Mathematics Senior High

丸つけているところの展開の仕方がわかりません!

・隣接3項間 基本 例題110 漸化式と極限 (2)、 00000 その条件によって褒められる数列 (c) の極限値を求めよ。 1 2=1, -(an+1+3an) 4 計方針は基本例題109と同じく,一般項an をnで表してから極限を求める 方般3項間漸化式でその支解をすると、そのとおいたの2次方程式 M ( 特性方程式) を解く。 その2解をα, βとすると、Bのとき の2通りに変形できる。 この変形を利用して解決する。 なお, 特性方程式の解に1を含むときは, 階差数列 が利用できる。 解答 与えられた漸化式を変形すると (1+1—an) an+2an+1 ゆえに, 数列{an+1-an} は初項1,公比 - - an+2)adn+1=β(an+1-Qan), an+2-Ban+1=0(a.ti-Ba.) an=a+ よって, n ≧2のとき 3\n-1 ²x = (-³) -¹ an+1_an= +(-3)*¹²* k=1\ k-1 よって n→∞ =0+ liman= 1-(-3)^²-² 1-(-³) 07 4 -lim-/-(1-(-3)^¹-¹) = 4 また a2-a=1-0=1 の等比数列で 1 3 4 n-1 -40-(-3)) したがって 注意 この問題のように, 単に数列{an}の極限を求めるときは, 2のときだけを考えてかまわない。つまり, n=1の ときの確認は必要ない。 n-11 別解 [am の求め方] 与えられた漸化式を変形すると 3 3 an+2an+1=- (an+1-an), an+2+ an+1=an+1+ 4 4 -7a₁-(-3) ³-²-1 an= P.176 まとめ 基本 109 3 4 a.- -/- (1-(-3)^"") an 3 4 025 -0.-(-3). am + fama+fa=1 ゆえに an+1-an=| -an = 3 an+1+ 4an=a₂+₁ 491=1 辺々引いて an =(x+3) を解くと 4x2=x+3 4x2-x-3=0 (x-1)(4x+3)=0 よって x=1, 3 4 {an}の階差数列{bn}が かれば,n≧2のとき n-1 an=a₁+Σbk k=1 18 Aa=1, B=- 極限を求めるとは, n→∞ の場合を考 -3/2 3 4' とα=- β= 場合の2通りで Man+1 を消去。

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English Junior High

問ニの②と問3、問4教えてください

3 次は,アメリカからの留学生のケイト (Kate) と, 高校生の勇太 (Yuta) との対話と, その日の夜に勇太が書いた日記の一部である。これらを読んで、後の各問に答えよ。 (One Monday morning. ) Kate Good morning, Yuta. Yuta : Good morning, Kate. You look sleepy. What's up? Kate: I went to bed late last night because I was talking a lot with my host family. Yuta Oh, I see. events. Kate: Some differences between America and Japan, for example, food, sports, and I think (find / we/ it's / to / interesting) differences in culture. Yuta: I think so, too. How about schools? Did you find any differences? Kate: Yes. For example, on the first day in Japan, I was surprised because my host brother walked to school by himself. In America, students - especially elementary school students go to school by school bus, or their parents drive them to school. Yuta: Really? In Japan, most students walk or ride a bike to school. I often get up late and ask my mother, "Drive me to school, please." But she says, "No! Run to school!" ja Kate: That's funny. I have another interesting story. Teachers here often say, "Take your textbooks home. Don't leave them at school." But students in America must leave them at school. Yuta Really? Why? Kate In America, students borrow textbooks from the states. We give them back at the end of the year. We (in/ not/them/ are/ write / must) or make them dirty. Yuta That's interesting! I want to know more about differences in culture. Kate Me, too. I want to talk about them with you. Do you have some time this weekend? Yuta I'm going to visit my grandfather's house on Sunday. But I'm ( f) on : Saturday. Please come to my house on Saturday. Kate OK. See you then. This morning I talked with Kate. She ( (1) ) me about some differences between American and Japanese schools. I was surprised that students in America [ 1. When I heard about that, I also became interested in differences in culture. So I'm going to talk about them with her on (@ ) at my house. 2

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