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数学 高校生

数学的帰納法の問題です。n=1とおいてa1を出すところまでは出来たのですが、n=kの時ではなくn<=kの時を考えるところが説明を読んでもよくわからないので解説お願いします。

数列{an} (ただしα > 0) について, 関係式 (a1+a2+....+an)=a+a2+......+an が成り立つとき, an=nであることを証明せよ。 3 指針 自然数nの問題であるから,数学的帰納法で証明する。 「n=kのときan=nが成り立つ」と仮定した場合, ak-1=k-1, ak-2=k-2, が 成り立つことを仮定していないこととなり, n=k+1のときについての次の等式 人が 作れなくなってしまう。 (1+2+......+k+ax+1)=1+2++k+αk+13 A したがって,n≦kの仮定が必要となる。 そこで,次の [1] [2] を示す数学的帰納法 を利用する。 下の検討も参照。 [1] n=1のとき成り立つ。 [2] n≦kのとき成り立つと仮定すると, n=k+1のときも成り立つ。 CHART 数学的帰納法 n≦kで成立を仮定する場合あり [1] n=1のとき,関係式から a2=0.3 解答 よって a2(a1-1)=0 α > 0から ゆえに, n=1のとき a =nは成り立つ。 <n=1のときの証明。 a=1 [2]n≦kのとき an=nが成り立つと仮定する。 n=k+1のときについて, 関係式から 3 {(1+2+......+k)+αk+1}=1+2°+....+k+ak+1 ... ① (①の左辺) = (1+2+... +k)+2(1+2+... +k) ak+1+ak+12 ² ={/12k(k+1) +2.1/2k(k+1)ax+x+ax+2 =13+23+......++k (k+1)ak+1+ak+12 ①の右辺と比較して ゆえに k(k+1)ak+1+ak+12=ak+13 ak+1 (ak+1+k){ak+1-(k+1)}=0 ak+1>0であるから ak+1=k+1 n≦kの仮定。 <n=k+1のときの 証明。 <a=1, a2=2, ak=k {ak+12-ak+1 -k(k+1)} =0 よって, n=k+1のときにも an=nは成り立つ。 [1], [2] から, すべての自然数nに対して α = n は成り立つ。 9

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英語 中学生

(4)合っていますか? 15行目くらいからだと思います

次の英文を読んで,(1)~(5)の問いに答えなさい。 Takashi visited Mr. Paul in London during spring vacation. famous places in London with Mr. Paul. He stayed at Mr. Paul's house. Takashi went to some One day, Takashi wanted to visit other places near London by himself and he told Mr. Paul about it. Mr. Paul said, "Go to Brighton. The city is very beautiful, so it's Takashi read the timetable many times and he (visit) by many people." station at s He looked at the clock in the planned to take a train at 8:40 in the morning. He arrived at the He sat on a chair and looked around him. Then he felt that something was wrong/ station building. It was 9:30. 8:30. But Takashi was very surprised, so he looked at his watch, but it was still 8:30. He found an old woman and asked, She looked at her watch and answered, "It's 8:30." He was relieved. suddenly, the old woman said to him again, "Oh, sorry. It's summer time now. 7.It started yesterday, so it's 9:30 10 "Excuse me, but what time is it now ?" now.' But just then her train came, so she stopped the conversation and ⑤( get) on the train. He went to Brighton. He enjoyed the city very much. Takashi didn't understand. took the next train at 9:40 and Takashi took a train back to London in the evening. He told Mr. Paul about his conversation with the old woman at the station. Mr. Paul laughed. Takashi asked, "What's summer time?" Mr. Paul said, "We have long daytime in summer. 15 From the end of March to the end of October, we put the clock forward an hour and then back again in fall. We do it to use the daytime more usefully. There are some good points, but also some problems." Takashi thought it was interesting. Mr. Paul said, "I want you to learn more about summer time." "I will," Takashi answered. After he came back to Japan, he went to the library and read a book about summer time.

解決済み 回答数: 1