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階差数列の解法 | 夢を叶える塾
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階差数列の問題の解法

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階差数列を利用して求める一般項や、その和を求める問題です。

基本問題

(1)1,2,5,10,17,...の数列の一般項を求めなさい
(2)1,1+3,1+3+3^2,1+3+3^2+3^3,...の数列の一般項を求めなさい

解き方

数列が等差でもなく、等比でもなく、ぱっと見て良くわからない数列のとき、階差を取りましょう
数列\{a_n\}は階差数列\{b_n\}を使ってn\geq 2の一般項を\displaystyle a_n=a_1+\sum_{k=1}^{n-1} b_kと表せます

解説

(1)1,2,5,10,17,...の数列の一般項を求めなさい
階差を見てみます。
元の数列\{a_n\}の階差数列\{b_n\}1,3,5,7,...で一般項2n-1のになっています。
数列\{b_n\}1,3,5,7,...で一般項2n-1のになっています。

数列\{a_n\}の一般項は、n\geq 2のとき\displaystyle a_n=a_1+\sum_{k=1}^{n-1} b_kでしたので、
\displaystyle a_n=1+\sum_{k=1}^{n-1} (2k-1)
\displaystyle a_n=1+2\cdot \frac{n(n-1)}{2}-(n-1)
\displaystyle a_n=1+n(n-1)-n+1
\displaystyle a_n=1+n^2-n-n+1
\displaystyle a_n=n^2-2n+2
これはn=1のときも成り立つので、すべてのnで一般項は\displaystyle a_n=n^2-2n+2

(2)1,1+3,1+3+3^2,1+3+3^2+3^3,...の数列の一般項を求めなさい
階差を見てみます。
元の数列\{a_n\}の階差数列\{b_n\}3,3^2,3^3,...です。
一般項は初項が3、公比が3の等比数列になっています。
\displaystyle b_n=3\cdot 3^{n-1}

数列\{a_n\}の一般項は、n\geq 2のとき\displaystyle a_n=a_1+\sum_{k=1}^{n-1} b_kでしたので、
\displaystyle a_n=1+\sum_{k=1}^{n-1} (3\cdot3^{k-1})
\displaystyle a_n=1+3\cdot\frac{3^{n-1}-1}{3-1}
\displaystyle a_n=\frac{2}{2}+\frac{3\cdot 3^{n-1}-3}{2}
\displaystyle a_n=\frac{1}{2}(3^n-1)
これはn=1のときも成り立つので、すべてのnで一般項は\displaystyle a_n=\frac{1}{2}(3^n-1)

応用問題

(1)1,1+3,1+3+3+3^2,1+3+3+3^2+3+3^2+3^3,...の数列の一般項を求めなさい
(2)2,4,7,13,24,42,...の数列の一般項を求めなさい

解き方

(1)は階差数列が少し複雑な時です。
基本問題の積み重ねになります。
(2)は階差がだめなら階差の階差です。
階差数列を階差数列の階差数列で求めます。

解説

(1)1,1+3,1+3+3+3^2,1+3+3+3^2+3+3^2+3^3,...の数列の一般項を求めなさい
ちょっと気持ち悪い感じがしますが、階差を見てみます。
元の数列\{a_n\}の階差数列\{b_n\}3,3+3^2,3+3^2+3^3,...です。
一般項は初項が3、公比が3の等比数列の和の形になっています。
\displaystyle b_n=3\frac{3^n-1}{3-1}=\frac{3}{2}(3^n-1)

数列\{a_n\}の一般項は、n\geq 2のとき\displaystyle a_n=a_1+\sum_{k=1}^{n-1} b_kでしたので、
\displaystyle a_n=1+\sum_{k=1}^{n-1} \frac{3}{2}(3^k-1)
ここからの計算でポイントがあります。
\displaystyle \sum_{k=1}^{n-1} 3^k
は、
\displaystyle \sum_{k=1}^{n-1} (3\cdot 3^{k-1})
として指数をk-1の形に変形して計算すると間違いが減ります
等比数列の和の公式はa\cdot r^{n-1}の形での和の公式だからです。
\displaystyle a_n=1+\sum_{k=1}^{n-1} \frac{3}{2}(3^k-1)
\displaystyle a_n=1+\sum_{k=1}^{n-1} \frac{3}{2}(3 \cdot 3^{k-1}-1)
\displaystyle a_n=1+ \frac{3}{2} \cdot 3\cdot \frac{3^{n-1}-1}{3-1}-\frac{3}{2}(n-1)
\displaystyle a_n=1+ \frac{3}{2} \cdot 3\cdot \frac{3^{n-1}-1}{2}-\frac{3}{2}n+\frac{3}{2}
\displaystyle a_n=\frac{4}{4}+ \frac{9}{4} 3^{n-1} - \frac{9}{4}-\frac{6}{4}n+\frac{6}{4}
\displaystyle a_n=\frac{9}{4} 3^{n-1} -\frac{6}{4}n + \frac{1}{4}
\displaystyle a_n=\frac{1}{4}\left(9\cdot 3^{n-1} -6n + 1\right)
これはn=1のときも成り立つので、すべてのnで一般項は\displaystyle a_n=\frac{1}{4}\left(9\cdot 3^{n-1} -6n + 1\right)

(2)2,4,7,13,24,42,...の数列の一般項を求めなさい
階差を見てみます。
2,3,6,11,18,...
まだよくわからない、という事で階差数列の階差数列を見てみましょう。
1,3,5,7,...という事で、2n-1の数列で表せそうです。

元の数列\{a_n\}の階差数列を\{b_n\}とします。
\{b_n\}の階差数列を\{c_n\}とします。

数列\{b_n\}の一般項は、n\geq 2のとき\displaystyle b_n=b_1+\sum_{k=1}^{n-1} c_kになります。
\displaystyle b_n=2+\sum_{k=1}^{n-1} (2k-1)
\displaystyle b_n=2+2\cdot \frac{n(n-1)}{2}-(n-1)
\displaystyle b_n=2+n(n-1)-n+1
\displaystyle b_n=2+n^2-n-n+1
\displaystyle b_n=n^2-2n+3
これはn=1のときも成り立つので、すべてのnで一般項は\displaystyle b_n=n^2-2n+3

数列\{a_n\}の一般項は、n\geq 2のとき\displaystyle a_n=a_1+\sum_{k=1}^{n-1} b_kになります。
\displaystyle a_n=2+\sum_{k=1}^{n-1} (k^2-2k+3)
\displaystyle a_n=2+\frac{n(n-1)(2n-1)}{6}-2\frac{n(n-1)}{2}+3(n-1)
\displaystyle a_n=2+\frac{n(2n^2-3n+1)}{6}-\frac{2n^2-2n}{2}+(3n-3)
\displaystyle a_n=\frac{12}{6}+\frac{2n^3-3n^2+n}{6}-\frac{6n^2-6n}{6}+\frac{18n-18)}{6}
\displaystyle a_n=\frac{12+2n^3-3n^2+n-6n^2+6n+18n-18}{6}
\displaystyle a_n=\frac{2n^3-9n^2+25n-18}{6}
\displaystyle a_n=\frac{1}{6}(2n^3-9n^2+25n-6)
これはn=1のときも成り立つので、すべてのnで一般項は\displaystyle a_n=\frac{1}{6}(2n^3-9n^2+25n-6)

終わりに

階差数列を利用して数列を求めるときはn \geq 2 に気を付けてください。
階差数列を利用した問題は、次の力が試されます。

  1. 階差数列の一般項を求めるという数列の一般項を求める力
  2. 階差数列から元の数列を計算するという階差数列の性質を利用する力
  3. 階差数列の和を計算するという和を求める力

今までの総復習ができるチャンスです!

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