If x-1-a1-a-a2-1-a1-1-a2- y-12a1-12a2-12a1-12a2-and z-1-3a1-1-3a2-1-3a1-1-3a-2-Then show that xy2-z2-2yz2-x2-3zx2-y2-0-

We can write x= 1 a _ 1 + 1 a _ 2 +x = 1 a _ 1 + 1 a _ 2 +x #160; ⇒ x= a _ 2 +x a _ 1 a _ 2 + a _ 1 x+1 ⇒ a _ 1 x ^ 2 + a _ 1 a _ 2 ...

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Byju's Answer

Standard XII

Mathematics

Global Maxima

If x=1/a1+a...

Question

If $x = \frac{1}{a_{1} +} \frac{a}{a_{2} +} \frac{1}{a_{1} +} \frac{1}{a_{2} +} . . . .,$ ,
$y = \frac{1}{2 a_{1} +} \frac{1}{2 a_{2} +} \frac{1}{2 a_{1} +} \frac{1}{2 a_{2} +} . . . . . .,$
and $z = \frac{1}{3 a_{1} +} \frac{1}{3 a_{2} +} \frac{1}{3 a_{1} +} \frac{1}{3 a - 2 +} . . . .,$ ,
Then show that $x (y^{2} - z^{2}) + 2 y (z^{2} - x^{2}) + 3 z (x^{2} - y^{2}) = 0$ .

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Solution

We can write $x = \frac{1}{a_{1} +} \frac{1}{a_{2} + x} = \frac{1}{a_{1} + \frac{1}{a_{2} + x}}$
$\Rightarrow x = \frac{a_{2} + x}{a_{1} a_{2} + a_{1} x + 1}$
$\Rightarrow a_{1} x^{2} + a_{1} a_{2} x - a_{2} = 0 ⟶ 1$
Similarly, $2 a_{1} y^{2} + 4 a_{1} a_{2} y - 2 a_{2} = 0 ⟶ 2$
$3 a_{1} z^{2} + 9 a_{1} a_{2} z - 3 a_{2} = 0 ⟶ 3$
From $1 & 2, 2 a_{1} (x^{2} - y^{2}) + 2 a_{1} a_{2} (x - 2 y) = 0$
From $1 & 3, 3 a_{1} (x^{2} - z^{2}) + 3 a_{1} a_{2} (x - 3 z) = 0$
Therefore, $\frac{2 (x^{2} - y^{2})}{3 (x^{2} - z^{2})} = \frac{2 (x - 2 y)}{3 (x - 3 z)}$
$\Rightarrow (x^{2} - y^{2}) (x - 3 z) = (x - 2 y) (x^{2} - z^{2})$
$\Rightarrow x^{3} - x y^{2} - 3 z x^{2} = x^{3} - x z^{2} - 2 y x^{2} + 2 y z^{2}$
$\Rightarrow x (z^{2} - y^{2}) + 2 y (x^{2} - z^{2}) + 3 z (y^{2} - x^{2}) = 0$
$\Rightarrow x (y^{2} - z^{2}) + 2 y (z^{2} - x^{2}) + 3 z (x^{2} - y^{2}) = 0$

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Similar questions

Q. If

(1 + x + x^{2} + . . . . . + x^{p})^{n} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . a_{n p} x^{n p}

Prove that

(1) a_{0} + a_{1} + a_{2} + . . . . . + a_{n p} = (p + 1)^{n}

(2) a_{1} + 2 a_{2} + 3 a_{3} + . . . . + n p . a_{n p} = \frac{1}{2} n p (p + 1)^{n}

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using mathematical induction.

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If x=1a1+aa2+1a1+1a2+....,,y=12a1+12a2+12a1+12a2+......,and z=13a1+13a2+13a1+13a−2+....,,Then show that x(y2−z2)+2y(z2−x2)+3z(x2−y2)=0.