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Rezolvati in R ecuatia :
x/1•2 + x/2•3 + x/3•4 +.....+ x/99•100 = 99/50


Răspuns :

[tex] \frac{x}{1*2}+ \frac{x}{2*3} +\frac{x}{3*4}+... \frac{x}{99*100}= \frac{99}{50}<=> \\ <=>x( \frac{1}{1*2}+ \frac{1}{2*3}+ \frac{1}{3*4}+...+ \frac{1}{99*100})= \frac{99}{50} <=> \\ <=>x(1- \frac{1}{2}+ \frac{1}{2}- \frac{1}{3}+ \frac{1}{3}- \frac{1}{4}+...+ \frac{1}{99}- \frac{1}{100})= \frac{99}{50}<=> \\ <=>x* \frac{99}{100}= \frac{99}{50}=> \\ \\ =>x=2. [/tex]
x ( [tex] \frac{1}{1*2} [/tex]+ [tex] \frac{1}{2*3} + \frac{1}{3*4} + ......+ \frac{1}{99*100} )= \frac{99}{50} [/tex]
x ( 1 - [tex]\frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ..... \frac{1}{99} - \frac{1}{100}) = \frac{99}{50} [/tex]
se reduc majoritatea, si ramane
x ( [tex]1- \frac{1}{100} ) = \frac{99}{50} [/tex] : amplifici 1 cu 100
[tex] \frac{99x}{100} = \frac{99}{50} [/tex]
se amplifica a doua fractie cu 2
[tex] <=> 99x= 99*2 <=> x=2[/tex]