An algebra problem by swastik p (5)

Algebra Level 3

Let x 1 , x 2 , ⋯ , x 2014 x_1,x_2, \cdots ,x_{2014} be real numbers other than 1 1 , such that x 1 + x 2 + ⋯ + x 2014 = 1 x_1+x_2+\cdots+x_{2014}=1 and x 1 1 − x 1 + x 2 1 − x 2 + ⋯ + x 2014 1 − x 2014 = 1 \dfrac{x_1}{1-x_1}+\dfrac{x_2}{1-x_2}+\cdots+\dfrac{x_{2014}}{1-x_{2014}}=1 .

What is the value of x 1 2 1 − x 1 + x 2 2 1 − x 2 + ⋯ + x 2014 2 1 − x 2014 \dfrac{x_1^2}{1-x_1}+\dfrac{x_2^2}{1-x_2}+\cdots+\dfrac{x_{2014}^2}{1-x_{2014}} ?

This is from Problems for everyone!!!


The answer is 0.

S P by s p , 16, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

S P
May 25, 2018

Observe that

∴ x 1 2 1 − x 1 + x 2 2 1 − x 2 + ⋯ + x 2014 2 1 − x 2014 = − ( x 1 − x 1 1 − x 1 + x 2 − x 2 1 − x 2 + ⋯ + x 2014 − x 2014 1 − x 2014 ) = − ( x 1 + x 2 + ⋯ + x 2014 − x 1 1 − x 1 − x 2 1 − x 2 ⋯ − x 2014 1 − x 2014 ) = − ( 1 − 1 ) = 0 \therefore \begin{aligned}\\& \dfrac{x_1^2}{1-x_1}+\dfrac{x_2^2}{1-x_2}+\cdots+\dfrac{x_{2014}^2}{1-x_{2014}} = -\left(x_1-\dfrac{x_1}{1-x_1}+x_2-\dfrac{x_2}{1-x_2}+\cdots +x_{2014}-\dfrac{x_{2014}}{1-x_{2014}}\right)\\& = -\left(x_1+x_2+\cdots +x_{2014}-\dfrac{x_1}{1-x_1}-\dfrac{x_2}{1-x_2}\cdots -\dfrac{x_{2014}}{1-x_{2014}}\right)\\& = -(1-1) \\& = \boxed{0} \end{aligned}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...