Matematické Fórum

stuart clark · 14. 06. 2014 06:30

If $\bf{x_{1}+x_{2}+x_{3}+............+x_{1007} = (2014)^2}$

and $\bf{\frac{x_{1}}{x_{1}+1} = \frac{x_{2}}{x_{2}+3} =.......... = \frac{x_{1007}}{x_{1007}+2013}}$ . Then value of $\bf{x_{253} = }$

sugyman · 14. 06. 2014 10:40

If we take the reciprocals of the fractions:
$\frac{x_{1}+1}{x_{1}} = \frac{x_{2}+3}{x_{2}} =.......... = \frac{x_{1007}+2013}{x_{1007}}$
Then substract 1:
$\frac{1}{x_{1}} = \frac{3}{x_{2}} =.......... = \frac{2013}{x_{1007}}$
and again inverse the fractions
$x_{1} = \frac{x_{2}}{3} =.......... = \frac{x_{1007}}{2013}$
We get:
$x_1+x_2+x_3+...+x_{1007}=x_1+3x_1+5x_1+...+2013x_1=x_1(1+3+5+...+2013)=x_1\cdot (1007)^2=2014^2$
$\Rightarrow x_1=4 \Rightarrow x_{253}=505\cdot x_1=2020$

stuart clark · 15. 06. 2014 10:30

Thanks ↑ sugyman:

Matematické Fórum

#1 14. 06. 2014 06:30

Sequence and Series

#2 14. 06. 2014 10:40

Re: Sequence and Series

#3 15. 06. 2014 10:30

Re: Sequence and Series

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