カテゴリー[ 昆虫| 田園| 花| 街| 数学・幾何学| 寺院| 城| 祭り| 鉄道| 海| 風力発電]

$\displaystyle x^n + \frac{1}{x^n}$ 型の対称式

2変数の対称式で y= 1/xと置くと、2変数の対称式のような振る舞いの式変形が可能です。

$\displaystyle x^n + \frac{1}{x^n}$ で $\displaystyle y = \frac{1}{x}$ と置くと、 $x^n + y^n$ となり、通常の2変数の対称式と見なせます。この場合、 $\displaystyle x y = x \cdot \frac{1}{x} = 1$ ， $\displaystyle x^2 y^2 = x^2 \cdot \frac{1}{x^2} = 1$ ， $\displaystyle x^3 y^3 = x^3 \cdot \frac{1}{x^3} = 1$ ，…， $\displaystyle x^n y^n = x^n \cdot \frac{1}{x^n} = 1$ となるため、 $\displaystyle x + \frac{1}{x}$ の値さえ判れば $\displaystyle x^n + \frac{1}{x^n}$ の値は必ず求めることができます。

2乗の和

$\displaystyle \left( x + \frac{1}{x} \right)^2 = x^2 + 2 x \cdot \frac{1}{x} + \frac{1}{x^2} \Leftrightarrow x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2$

3乗の和

$\displaystyle \left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3 x \cdot \frac{1}{x} \cdot \left( x + \frac{1}{x} \right) \\ \displaystyle \Leftrightarrow x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right) = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\}$

$\displaystyle \left( x + \frac{1}{x} \right) \left( x^2 + \frac{1}{x^2} \right) = x^3 + x + \frac{1}{x} + \frac{1}{x^3} = x^3 + \frac{1}{x^3} + \left( x + \frac{1}{x} \right) \\ \displaystyle \Leftrightarrow x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) = \left( x + \frac{1}{x} \right) \left\{ \left( x^2 + \frac{1}{x^2} \right) - 1 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 2 - 1 \right\} = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\}$

4乗の和

$\displaystyle \left( x + \frac{1}{x} \right)^4 = x^4 + \frac{1}{x^4} + 4 \left( x^2 + \frac{1}{x^2}\right) + 6 \\ \displaystyle \Leftrightarrow x^4 + \frac{1}{x^4} = \left( x + \frac{1}{x} \right)^4 - 4 \left( x^2 + \frac{1}{x^2}\right) -6 = \left( x + \frac{1}{x} \right)^4 - 4 \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} -6 \\ \displaystyle = \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 8 - 6 = \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 2$

$\displaystyle \left( x^2 + \frac{1}{x^2}\right)^2 = x^4 + 2 x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} \\ \displaystyle \Leftrightarrow x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2}\right)^2 -2 = \left\{ \left( x + \frac{1}{x}\right)^2 - 2 \right\}^2 - 2 = \left( x + \frac{1}{x}\right)^4 - 4 \left( x + \frac{1}{x}\right)^2 + 4 - 2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^4 - 4 \left( x + \frac{1}{x}\right)^2 + 2$

$\displaystyle x^4 + \frac{1}{x^4} = \left( x^2 \right)^2 + \frac{1}{\left( x^2 \right)^2 } = \left( x^2 + \frac{1}{x^2}\right)^2 -2 = \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\}^2 - 2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^4 - 4 \left( x + \frac{1}{x}\right)^2 + 4 - 2 = \left( x + \frac{1}{x}\right)^4 - 4 \left( x + \frac{1}{x}\right)^2 + 2$

$\displaystyle \left( x + \frac{1}{x} \right) \left( x^3 + \frac{1}{x^3} \right) = x^4 + x^2 + \frac{1}{x^2} + \frac{1}{x^4} = x^4 + \frac{1}{x^4} + \left( x^2 + \frac{1}{x^2} \right) \\ \displaystyle \Leftrightarrow x^4 + \frac{1}{x^4} = \left( x + \frac{1}{x} \right) \left( x^3 + \frac{1}{x^3} \right) - \left( x^2 + \frac{1}{x^2} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right) \left[ \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \right] - \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right)^2 \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} - \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right)^4 - 3 \left( x + \frac{1}{x} \right)^2 - \left( x + \frac{1}{x} \right)^2 + 2 = \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 2$

5乗の和

$\displaystyle \left( x + \frac{1}{x} \right)^5 = x^5 + \frac{1}{x^5} + 5 \left( x^3 + \frac{1}{x^3}\right) + 10 \left( x + \frac{1}{x} \right) \\ \displaystyle \Leftrightarrow x^5 + \frac{1}{x^5} = \left( x + \frac{1}{x} \right)^5 - 5 \left( x^3 + \frac{1}{x^3}\right) - 10 \left( x + \frac{1}{x} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right)^5 - 5 \left\{ \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right) \right\} - 10 \left( x + \frac{1}{x} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right) \left[ \left( x + \frac{1}{x} \right)^4 - 5 \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} - 10 \right] = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 -5 \left( x + \frac{1}{x} \right)^2 + 15 - 10 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 -5 \left( x + \frac{1}{x} \right)^2 + 5 \right\}$

$\displaystyle \left( x^3 + \frac{1}{x^3}\right) \left( x^2 + \frac{1}{x^2}\right) = x^5 + \frac{1}{x^5} + \left( x + \frac{1}{x} \right) \\ \displaystyle \Leftrightarrow x^5 + \frac{1}{x^5} = \left( x^3 + \frac{1}{x^3}\right) \left( x^2 + \frac{1}{x^2}\right) - \left( x + \frac{1}{x} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} - \left( x + \frac{1}{x} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right) \left[ \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} - 1 \right] \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 - 5 \left( x + \frac{1}{x} \right)^2 + 6 - 1 \right\} = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 - 5 \left( x + \frac{1}{x} \right)^2 + 5 \right\}$

$\displaystyle \left( x + \frac{1}{x} \right) \left( x^4 + \frac{1}{x^4} \right) = x^5 + x^3 + \frac{1}{x^3} + \frac{1}{x^5} = x^5 + \frac{1}{x^5} + \left( x^3 + \frac{1}{x^3} \right) \\ \displaystyle \Leftrightarrow x^5 + \frac{1}{x^5} = \left( x + \frac{1}{x} \right) \left( x^4 + \frac{1}{x^4} \right) - \left( x^3 + \frac{1}{x^3} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 2 \right\} - \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 2 - \left( x + \frac{1}{x} \right)^2 + 3 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 - 5 \left( x + \frac{1}{x} \right)^2 + 5 \right\}$

6乗の和

$\displaystyle x^6 + \frac{1}{x^6} = \left( x^2 \right)^3 + \frac{1}{\left( x^2 \right)^3} = \left( x^2 + \frac{1}{x^2} \right) \left\{ \left( x^2 + \frac{1}{x^2} \right)^2 - 3 \right\} \\ \displaystyle = \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} \left[ \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\}^2 - 3 \right] = \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} \left\{ \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 4 - 3 \right\} \\ \displaystyle = \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} \left\{ \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 1 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right)^6 - 4 \left( x + \frac{1}{x} \right)^4 + \left( x + \frac{1}{x} \right)^2 - 2 \left( x + \frac{1}{x} \right)^4 + 8 \left( x + \frac{1}{x} \right)^2 - 2 \\ \displaystyle = \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 2$

$\displaystyle \left( x + \frac{1}{x} \right)^6 = x^6 + \frac{1}{x^6} + 6 \left( x^4 + \frac{1}{x^4} \right) + 15 \left( x^2 + \frac{1}{x^2} \right) + 20 \\ \displaystyle = x^6 + \frac{1}{x^6} + 6 \left\{ \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 2 \right\} + 15 \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} + 20 \\ \displaystyle = x^6 + \frac{1}{x^6} + 6 \left( x + \frac{1}{x} \right)^4 - 24 \left( x + \frac{1}{x} \right)^2 + 12 + 15 \left( x + \frac{1}{x} \right)^2 - 30 + 20 \\ \displaystyle = x^6 + \frac{1}{x^6} + 6 \left( x + \frac{1}{x} \right)^4 - 9 \left( x + \frac{1}{x} \right)^2 + 2 \\ \displaystyle \Leftrightarrow x^6 + \frac{1}{x^6} = \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 2$

7乗の和

$\displaystyle \left( x + \frac{1}{x} \right)^7 = x^7 + \frac{1}{x^7} + 7 \left( x^5 + \frac{1}{x^5} \right) + 21 \left( x^3 + \frac{1}{x^3} \right) + 35 \left( x + \frac{1}{x} \right) \\ \displaystyle \Leftrightarrow x^7 + \frac{1}{x^7} = \left( x + \frac{1}{x} \right)^7 - 7 \left( x^5 + \frac{1}{x^5} \right) - 21 \left( x^3 + \frac{1}{x^3} \right) - 35 \left( x + \frac{1}{x} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right)^7 - 7 \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 -5 \left( x + \frac{1}{x} \right)^2 + 5 \right\} - 21 \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} - 35 \left( x + \frac{1}{x} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right) \left[ \left( x + \frac{1}{x} \right)^6 - 7 \left\{ \left( x + \frac{1}{x} \right)^4 -5 \left( x + \frac{1}{x} \right)^2 + 5 \right\} - 21 \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} - 35 \right] \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^6 - 7 \left( x + \frac{1}{x} \right)^4 + 35 \left( x + \frac{1}{x} \right)^2 - 35 - 21 \left( x + \frac{1}{x} \right)^2 + 63 - 35 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^6 - 7 \left( x + \frac{1}{x} \right)^4 + 14 \left( x + \frac{1}{x} \right)^2 - 7 \right\}$

$\displaystyle \left( x + \frac{1}{x} \right) \left( x^6 + \frac{1}{x^6} \right) = x^7 + \frac{1}{x^5} + x^5 + \frac{1}{x^7} = x^7 + \frac{1}{x^7} + \left( x^5 + \frac{1}{x^5} \right) \\ \displaystyle \Leftrightarrow x^7 + \frac{1}{x^7} = \left( x + \frac{1}{x} \right) \left( x^6 + \frac{1}{x^6} \right) - \left( x^5 + \frac{1}{x^5} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 2 \right\} - \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 - 5 \left( x + \frac{1}{x} \right)^2 + 5 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 2 - \left( x + \frac{1}{x} \right)^4 + 5 \left( x + \frac{1}{x} \right)^2 - 5 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^6 - 7 \left( x + \frac{1}{x} \right)^4 + 14 \left( x + \frac{1}{x} \right)^2 - 7 \right\}$

8乗の和

$\displaystyle x^8 + \frac{1}{x^8} = \left( x^4 \right)^2 + \frac{1}{\left( x^4 \right)^2 } = \left( x^4 + \frac{1}{x^4}\right)^2 -2 = \left\{ \left( x + \frac{1}{x}\right)^4 - 4 \left( x + \frac{1}{x}\right)^2 + 2 \right\}^2 - 2 \\ \scriptsize \displaystyle = \left\{ \left( x + \frac{1}{x}\right)^4 \right\}^2 + \left\{ - 4 \left( x + \frac{1}{x}\right)^2 \right\}^2 + 2^2 + 2 \left( x + \frac{1}{x}\right)^4 \left\{ - 4 \left( x + \frac{1}{x}\right)^2 \right\} + 2 \left\{ - 4 \left( x + \frac{1}{x}\right)^2 \right\} \cdot 2 + 2 \cdot 2 \cdot \left( x + \frac{1}{x}\right)^4 -2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^8 + 16 \left( x + \frac{1}{x}\right)^4 + 4 - 8 \left( x + \frac{1}{x}\right)^6 - 16 \left( x + \frac{1}{x}\right)^2 + 4 \left( x + \frac{1}{x}\right)^4 - 2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^8 - 8 \left( x + \frac{1}{x}\right)^6 + 20 \left( x + \frac{1}{x}\right)^4 - 16 \left( x + \frac{1}{x}\right)^2 + 2$

$\displaystyle \left( x + \frac{1}{x} \right)^8 = x^8 + \frac{1}{x^8} + 8 \left( x^6 + \frac{1}{x^6} \right) + 28 \left( x^4 + \frac{1}{x^4} \right) + 56 \left( x^2 + \frac{1}{x^2} \right) + 70 \\ \displaystyle \Leftrightarrow x^8 + \frac{1}{x^8} = \left( x + \frac{1}{x} \right)^8 - 8 \left( x^6 + \frac{1}{x^6} \right) - 28 \left( x^4 + \frac{1}{x^4} \right) - 56 \left( x^2 + \frac{1}{x^2} \right) - 70 \\ \displaystyle = \left( x + \frac{1}{x} \right)^8 - 8 \left\{ \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 2 \right\} - 28 \left\{ \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 2 \right\} \\ \displaystyle - 56 \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} - 70 \\ \displaystyle = \left( x + \frac{1}{x}\right)^8 - 8 \left( x + \frac{1}{x}\right)^6 + 48 \left( x + \frac{1}{x}\right)^4 - 72 \left( x + \frac{1}{x}\right)^2 + 16 - 28 \left( x + \frac{1}{x}\right)^4 + 112 \left( x + \frac{1}{x}\right)^2 - 56 \\ \displaystyle - 56 \left( x + \frac{1}{x}\right)^2 + 112 - 70 \\ \displaystyle = \left( x + \frac{1}{x}\right)^8 - 8 \left( x + \frac{1}{x}\right)^6 + 20 \left( x + \frac{1}{x}\right)^4 - 16 \left( x + \frac{1}{x}\right)^2 + 2$

$\displaystyle \left( x + \frac{1}{x} \right) \left( x^7 + \frac{1}{x^7} \right) = x^8 + \frac{1}{x^6} + x^6 + \frac{1}{x^8} = x^8 + \frac{1}{x^8} + \left( x^6 + \frac{1}{x^6} \right) \\ \displaystyle \Leftrightarrow x^8 + \frac{1}{x^8} = \left( x + \frac{1}{x} \right) \left( x^7 + \frac{1}{x^7} \right) - \left( x^6 + \frac{1}{x^6} \right) \\ \displaystyle = \left( x + \frac{1}{x} \right) \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^6 - 7 \left( x + \frac{1}{x} \right)^4 + 14 \left( x + \frac{1}{x} \right)^2 - 7 \right\} \\ \displaystyle - \left\{ \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 2 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right)^2 \left\{ \left( x + \frac{1}{x} \right)^6 - 7 \left( x + \frac{1}{x} \right)^4 + 14 \left( x + \frac{1}{x} \right)^2 - 7 \right\} \\ \displaystyle - \left\{ \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 2 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right)^8 - 7 \left( x + \frac{1}{x} \right)^6 + 14 \left( x + \frac{1}{x} \right)^4 - 7 \left( x + \frac{1}{x} \right)^2 - \left( x + \frac{1}{x} \right)^6 + 6 \left( x + \frac{1}{x} \right)^4 - 16 \left( x + \frac{1}{x} \right)^2 + 2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^8 - 8 \left( x + \frac{1}{x}\right)^6 + 20 \left( x + \frac{1}{x}\right)^4 - 16 \left( x + \frac{1}{x}\right)^2 + 2$

9乗の和

$\displaystyle \left( x + \frac{1}{x} \right)^9 = x^9 + \frac{1}{x^9} + 9 \left( x^7 + \frac{1}{x^7} \right) + 36 \left( x^5 + \frac{1}{x^5} \right) + 84 \left( x^3 + \frac{1}{x^3} \right) + 126 \left( x + \frac{1}{x} \right) \\ \displaystyle \Leftrightarrow x^9 + \frac{1}{x^9} = \left( x + \frac{1}{x} \right)^9 - 9 \left( x^7 + \frac{1}{x^7} \right) - 36 \left( x^5 + \frac{1}{x^5} \right) - 84 \left( x^3 + \frac{1}{x^3} \right) - 126 \left( x + \frac{1}{x} \right) \\ \scriptsize \displaystyle = \left( x + \frac{1}{x} \right)^9 - 9 \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^6 - 7 \left( x + \frac{1}{x} \right)^4 + 14 \left( x + \frac{1}{x} \right)^2 - 7 \right\} - 36 \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^4 -5 \left( x + \frac{1}{x} \right)^2 + 5 \right\} \\ \scriptsize \displaystyle - 84 \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} - 126 \left( x + \frac{1}{x} \right) \\ \scriptsize \displaystyle = \left( x + \frac{1}{x} \right) \left[ \left( x + \frac{1}{x} \right)^8 - 9 \left\{ \left( x + \frac{1}{x} \right)^6 - 7 \left( x + \frac{1}{x} \right)^4 + 14 \left( x + \frac{1}{x} \right)^2 - 7 \right\} - 36 \left\{ \left( x + \frac{1}{x} \right)^4 -5 \left( x + \frac{1}{x} \right)^2 + 5 \right\} - 84 \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} - 126 \right] \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^8 - 9 \left( x + \frac{1}{x} \right)^6 + 27 \left( x + \frac{1}{x} \right)^4 - 30 \left( x + \frac{1}{x} \right)^2 - 9 \right\}$

$\displaystyle x^{9} + \frac{1}{x^{9}} = \left( x^3 \right)^3 + \frac{1}{\left( x^3 \right)^3} = \left( x^3 + \frac{1}{x^3} \right) \left\{ \left( x^3 + \frac{1}{x^3} \right)^2 - 3 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \left[ \left[ \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \right]^2 - 3 \right] \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \left[ \left( x + \frac{1}{x} \right)^2 \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\}^2 - 3 \right] \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \left[ \left( x + \frac{1}{x} \right)^2 \left\{ \left( x + \frac{1}{x} \right)^4 - 6 \left( x + \frac{1}{x} \right)^2 + 9 \right\} - 3 \right] \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^2 - 3 \right\} \left\{ \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 3 \right\} \\ \small \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^8 - 6 \left( x + \frac{1}{x} \right)^6 + 9 \left( x + \frac{1}{x} \right)^4 - 3 \left( x + \frac{1}{x} \right)^2 -3 \left( x + \frac{1}{x} \right)^6 + 18 \left( x + \frac{1}{x} \right)^4 - 27 \left( x + \frac{1}{x} \right)^2 - 9 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^8 - 9 \left( x + \frac{1}{x} \right)^6 + 27 \left( x + \frac{1}{x} \right)^4 - 30 \left( x + \frac{1}{x} \right)^2 - 9 \right\}$

$\displaystyle \left( x + \frac{1}{x} \right) \left( x^8 + \frac{1}{x^8} \right) = x^9 + \frac{1}{x^7} + x^7 + \frac{1}{x^9} = x^9 + \frac{1}{x^9} + \left( x^7 + \frac{1}{x^7} \right) \\ \displaystyle \Leftrightarrow x^9 + \frac{1}{x^9} = \left( x + \frac{1}{x} \right) \left( x^8 + \frac{1}{x^8} \right) - \left( x^7 + \frac{1}{x^7} \right) \\ \scriptsize \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x}\right)^8 - 8 \left( x + \frac{1}{x}\right)^6 + 20 \left( x + \frac{1}{x}\right)^4 - 16 \left( x + \frac{1}{x}\right)^2 + 2 \right\} - \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x} \right)^6 - 7 \left( x + \frac{1}{x} \right)^4 + 14 \left( x + \frac{1}{x} \right)^2 - 7 \right\} \\ \scriptsize \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x}\right)^8 - 8 \left( x + \frac{1}{x}\right)^6 + 20 \left( x + \frac{1}{x}\right)^4 - 16 \left( x + \frac{1}{x}\right)^2 + 2 - \left( x + \frac{1}{x} \right)^6 + 7 \left( x + \frac{1}{x} \right)^4 - 14 \left( x + \frac{1}{x} \right)^2 + 7 \right\} \\ \displaystyle = \left( x + \frac{1}{x} \right) \left\{ \left( x + \frac{1}{x}\right)^8 - 9 \left( x + \frac{1}{x}\right)^6 + 27 \left( x + \frac{1}{x}\right)^4 - 30 \left( x + \frac{1}{x}\right)^2 + 2 \right\}$

10乗の和

$\displaystyle x^{10} + \frac{1}{x^{10}} = \left( x^5 \right)^2 + \frac{1}{\left( x^5 \right)^2 } = \left( x^5 + \frac{1}{x^5}\right)^2 -2 = \left[ \left( x + \frac{1}{x}\right) \left\{ \left( x + \frac{1}{x}\right)^4 - 5 \left( x + \frac{1}{x}\right)^2 + 5 \right\} \right]^2 - 2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^2 \left\{ \left( x + \frac{1}{x}\right)^4 - 5 \left( x + \frac{1}{x}\right)^2 + 5 \right\}^2 - 2 \\ \scriptsize \displaystyle = \left( x + \frac{1}{x}\right)^2 \left[ \left\{ \left( x + \frac{1}{x}\right)^4 \right\}^2 + \left\{ - 5 \left( x + \frac{1}{x}\right)^2 \right\}^2 + 5^2 + 2 \left( x + \frac{1}{x}\right)^4 \cdot \left\{ - 5 \left( x + \frac{1}{x}\right)^2 \right\} + 2 \left\{ - 5 \left( x + \frac{1}{x}\right)^2 \right\} \cdot 5 + 2 \cdot 5 \cdot \left( x + \frac{1}{x}\right)^4 \right] - 2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^2 \left\{ \left( x + \frac{1}{x}\right)^8 + 25 \left( x + \frac{1}{x}\right)^4 + 25 - 10 \left( x + \frac{1}{x}\right)^6 - 50 \left( x + \frac{1}{x}\right)^2 + 10 \left( x + \frac{1}{x}\right)^4 \right\} - 2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^2 \left\{ \left( x + \frac{1}{x}\right)^8 - 10 \left( x + \frac{1}{x}\right)^6 + 35 \left( x + \frac{1}{x}\right)^4 - 50 \left( x + \frac{1}{x}\right)^2 + 25 \right\} -2 \\ \displaystyle = \left( x + \frac{1}{x}\right)^{10} - 10 \left( x + \frac{1}{x}\right)^8 + 35 \left( x + \frac{1}{x}\right)^6 - 50 \left( x + \frac{1}{x}\right)^4 + 25 \left( x + \frac{1}{x}\right)^2 - 2$

$\displaystyle \left( x + \frac{1}{x} \right)^{10} = x^{10} + \frac{1}{x^{10}} + 10 \left( x^8 + \frac{1}{x^8} \right) + 45 \left( x^6 + \frac{1}{x^6} \right) + 120 \left( x^4 + \frac{1}{x^4} \right) + 210 \left( x^2 + \frac{1}{x^2} \right) + 252 \\ \displaystyle \Leftrightarrow x^{10} + \frac{1}{x^{10}} = \left( x + \frac{1}{x} \right)^{10} - 10 \left( x^8 + \frac{1}{x^8} \right) - 45 \left( x^6 + \frac{1}{x^6} \right) - 120 \left( x^4 + \frac{1}{x^4} \right) - 210 \left( x^2 + \frac{1}{x^2} \right) - 252 \\ \displaystyle = \left( x + \frac{1}{x} \right)^{10} - 10 \left\{ \left( x + \frac{1}{x}\right)^8 - 8 \left( x + \frac{1}{x}\right)^6 + 20 \left( x + \frac{1}{x}\right)^4 - 16 \left( x + \frac{1}{x}\right)^2 + 2 \right\} \\ \displaystyle - 45 \left\{ \left( x + \frac{1}{x} \right)^6 - 6 \left( x + \frac{1}{x} \right)^4 + 9 \left( x + \frac{1}{x} \right)^2 - 2 \right\} - 120 \left\{ \left( x + \frac{1}{x} \right)^4 - 4 \left( x + \frac{1}{x} \right)^2 + 2 \right\} \\ \displaystyle - 210 \left\{ \left( x + \frac{1}{x} \right)^2 - 2 \right\} - 252 \\ \displaystyle = \left( x + \frac{1}{x} \right)^{10} - 10 \left( x + \frac{1}{x} \right)^{8} + \left( 80 - 45 \right) \left( x + \frac{1}{x} \right)^{6} - \left( 200 - 270 + 120 \right) \left( x + \frac{1}{x} \right)^{4} \\ \displaystyle + \left( 160 - 405 + 480 - 210 \right) \left( x + \frac{1}{x} \right)^{2} - \left( 20 - 90 + 240 - 420 + 252 \right) \\ \displaystyle = \left( x + \frac{1}{x}\right)^{10} - 10 \left( x + \frac{1}{x}\right)^8 + 35 \left( x + \frac{1}{x}\right)^6 - 50 \left( x + \frac{1}{x}\right)^4 + 25 \left( x + \frac{1}{x}\right)^2 - 2$

CANADA'S WINDVIEW

趣味の写真を投稿していきます。昆虫好きな長男と一緒に昆虫を追いかけています。最初の年はセミやカマキリ、次の年はカブトムシ、トンボ、そして今年は…

(x^n)+1/(x^n)型の対称式