Страница 334 номер 1100, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Решить уравнение: 1) \(\frac{P_{x+1}}{P_{x-1}} = 30\); 2) \(\frac{P_x}{P_{x-2}} = 42\); 3) \(\frac{1}{P_{x-5}} = \frac{56}{P_{x-3}}\); 4) \(\frac{1}{P_{x-4}} = \frac{110}{P_{x-2}}\); 5) \(A_{x+1}^3 = 72(x - 1)\); 6) \(A_{x-1}^4 = 40(x - 2)(x - 3)\); 7) \(5C_{n+1}^3 = 8C_n^4\); 8) \(C_n^3 = 4C_{n-2}^2\).
1) \(\frac{P_{x + 1}}{P_{x - 1}} = 30\)
\(x - 1 \geqslant 0, x \geqslant 1;\)
\(\dfrac{(x + 1)!}{(x - 1)!} = 30,\)
\(x(x + 1) = 30,\)
\(x^{2} + x - 30 = 0,\)
\(x_{1{,}2} = \dfrac{-1 \pm \sqrt{1 + 120}}{2},\)
\(x_{1} = 5\), \(x_{2} = - 6\) – не подходит;
2) \(\frac{P_{x}}{P_{x - 2}} = 42\)
\(x - 2 \geqslant 0, x \geqslant 2;\)
\(\dfrac{x!}{(x - 2)!} = 42,\)
\(x(x - 1) = 42,\)
\(x^{2} - x - 42 = 0,\)
\(x_{1{,}2} = \dfrac{1 \pm \sqrt{1 + 168}}{2},\)
\(x_{1} = 7\), \(x_{2} = - 6\) – не подходит;
3) \(\frac{1}{P_{x - 5}} = \frac{56}{P_{x - 3}}\)
\(x - 5 \geqslant 0, x \geqslant 5;\)
\(\dfrac{(x - 3)!}{(x - 5)!} = 56,\)
\((x - 3)(x - 4) = 56,\)
\(x^{2} - 7x - 44 = 0,\)
\(x_{1{,}2} = \dfrac{7 \pm \sqrt{49 + 176}}{2},\)
\(x_{1} = 11\), \(x_{2} = - 4\) – не подходит;
4) \(\frac{1}{P_{x - 4}} = \frac{110}{P_{x - 2}}\)
\(x - 4 \geqslant 0, x \geqslant 4;\)
\(\dfrac{(x - 2)!}{(x - 4)!} = 110,\)
\((x - 3)(x - 2) = 110,\)
\(x^{2} - 5x - 104 = 0,\)
\(x_{1{,}2} = \dfrac{5 \pm \sqrt{25 + 416}}{2},\)
\(x_{1} = 13\), \(x_{2} = - 8\) – не подходит;
5) \(A_{x + 1}^{3} = 72(x - 1)\)
\(\dfrac{(x + 1)!}{(x + 1 - 3)!} = 72(x - 1),\)
\(x(x - 1)(x + 1) = 72(x - 1),\)
\(x^{2} + x - 72 = 0,\)
\(x_{1{,}2} = \dfrac{-1 \pm \sqrt{1 + 288}}{2},\)
\(x_{1} = 8\), \(x_{2} = - 9\) – не подходит;
6) \(A_{x - 1}^{4} = 40(x - 2)(x - 3)\)
\(\dfrac{(x - 1)!}{(x - 1 - 4)!} = 40(x - 2)(x - 3),\)
\((x - 4)(x - 3)(x - 2)(x - 1) = 40(x - 2)(x - 3),\)
\(x^{2} - 5x - 36 = 0,\)
\(x_{1{,}2} = \dfrac{5 \pm \sqrt{25 + 144}}{2},\)
\(x_{1} = 9\), \(x_{2} = - 4\) – не подходит;
7) \(5C_{n + 1}^{3} = 8C_{n}^{4}\)
\(5\dfrac{(n + 1)!}{3!(n + 1 - 3)!} = 8\dfrac{n!}{4!(n - 4)!},\)
\((n - 1)(n + 1)n = \dfrac{8}{5} \cdot \dfrac{3!}{4!}(n - 3)(n - 2)(n - 1)n.\)
\(n + 1 = \dfrac{2}{5}\left( n^{2} - 5n + 6 \right),\)
\(2n^{2} - 15n + 7 = 0,\)
\(n_{1{,}2} = \dfrac{15 \pm \sqrt{225 - 56}}{4},\)
\(n_{1} = 7\), \(n_{2} = \frac{1}{2}\) – не подходит;
8) \(C_{n}^{3} = 4C_{n - 2}^{2}\)
\(\dfrac{n!}{3!(n - 3)!} = 4\dfrac{(n - 2)!}{2!(n - 4)!},\)
\(n(n - 1)(n - 2) = 4 \cdot 3(n - 3)(n - 2),\)
\(n^{2} - n = 12n - 36,\)
\(n^{2} - 13n + 36 = 0,\)
\(n_{1{,}2} = \dfrac{13 \pm \sqrt{169 - 144}}{2},\)
\(n_{1} = 9, n_{2} = 4.\)
