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== 等比数列の和 ==
《解説》
○ 正の整数の和
k=1nk=n(n+1)2
や正の整数の2乗の和や3乗の和
k=1nk2=n(n+1)(2n+1)6
k=1nk3={n(n+1)2}2
の公式を使うときは,どちらかと言えば機械的に公式を適用すれば答が得られます.

○ しかし,等比数列の和を表しているΣ記号から結果を求めるには,上のような機械的な当てはめだけでは無理で,
初項a公比r項数nの3要素に分けて読み取り,「等比数列の和の公式」
…(*)
に代入するとできます.

○ 等比数列の和といっても,
k=143×2k1
のように,初項3公比2項数4が見ただけですぐ分かる形(an=a r n−1)になったものばかりとは限りません.
k=1n2k
k=1n3×2k+1
など,さまざまな形で書かれる等比数列の和を求めるには,
(1) 初項a公比r項数nの3要素を「分けて読み取る」

(2) 「上の公式(*)に代入する」
という操作に慣れなければなりません.

【例1】 次の和を求めてください.
k=1n2k

(答案)
<等比数列の3要素を読み取る>
  • k=1を代入:a=a1=2
  • k123...
    ak248...
    2倍ずつになっているから公比r=2
  • 1からnだから項数n

に代入する.


.2(2n−1)2−1nnnnnnn=2(2n−1) …(答)
2n+1−2 と書いてもよい
(別解)
k=1n2×2k1
と変形して,初項a=2公比r=2項数nを読み取ってもよいが,この教材では数学が不得意な人を念頭に置いているので,このような器用な変形は薦めない.
特に,an=a r n−1n−1が嫌な形をしており,覚え
にくいと考えられる.

【例2】 次の和を求めてください.
k=2n3×4k+1

(答案)
<等比数列の3要素を読み取る>
  • k=2を代入:a=3×43=192
    例えば,3×22は,62にはならない.
    このような「掛け算」と「累乗」がある式では,必ず累乗の計算を優先的に行い,できあがった結果に掛け算を行うので
    3×4=12になります.
    同様にして,3×42=122=144×
    3×42=3×16=48
    同様にして,3×43=123=1728×
    3×43=3×64=192
  • k234...
    ak1927683072...
    4倍ずつになっているから公比r=4
  • 2からnだから(1からnでn個.これよりも1つ少ない)項数n−1

に代入する.


.192(4n−1−1)4−1nnnnnnnnn=.192(4n−1−1)3nnnnnnnnn=64(4n−1−1) …(答)

【例3】 次の和を求めてください.
k=0n3k1

(答案)
<等比数列の3要素を読み取る>
  • k=0を代入:a=3−1=.13n
    数列では,k=1, 2, 3, ..を使った
    a1 , a2 , a3 , ...が最もよく使われますが,
    k=0, 1, 2, 3, ..を使った
    a0 , a1 , a2 , a3 , ...も使います.この場合は,a0が初項になります.
  • k012...
    ak.13n13...
    3倍ずつになっているから公比r=3
  • 0からnだから(1からnでn個.これよりも1つ多い)項数n+1
    3k−1の形から,項数n−1などと考えてはいけない.
    項数は,一般項の式とは関係なく決まり,kの値の幾らから幾らまで使うかだけで決まる.(Σ記号の「下に書かれた数字」から「上に書かれた数字」まで何個あるのかということ)

に代入する.


.13n.3n+1−13−1nnnnnn=.3n+1−16nnnnnn …(答)

《問題》
○左辺に等しいものを右辺から見つけなさい.
○左辺から1つ選び,続けて右辺から1つ選びなさい.合っていれば消えます.間違えば消えません.
○間違った場合は,下にHELPボタンが出ます.HELPを読む場合でも読まない場合でも,新たに問題を選択すれば解答を再開できます.








































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■[個別の頁からの質問に対する回答][Σ等比数列について/17.7.22]
等比数列の和の公式について質問させてください。 先生のページでは、項比rから-1するという形になっていますが、 別の書籍等では、1から項比rをマイナスするという形になっているものもあります。 この違いは何に起因するのでしょうか? ご教示ください。
=>[作者]:連絡ありがとう.



とは等しいので,どちらで表してもかまいません.(分母と分子の両方に -1を掛ける[全体には1を掛けることになる]と分数としては等しい)
あえて言えば,生徒が自然に感じる(したがって,計算間違いしにくい)方を使うようにしているだけでしょう.
数学ⅡBの段階では有限数列しか登場しないから,和の公式からさらに変形する場合でも符号を間違わないように最高次の係数を正にして

が好まれるでしょう.これに対して,数学Ⅲでは無限等比級数の和も扱うので
のとき
を使って

とする方が自然に見えます.

でももちろん「正しい」のですが,こちらを好む人は少ないでしょう.
■[個別の頁からの質問に対する回答][等比数列の和について/17.6.14]
素晴らしい よく理解できた 練習問題のなんいとまをもう少しあげてもいいと思う
=>[作者]:連絡ありがとう.さらに進んだものはこの頁など

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