Question

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

Answer 1

$\frac{x}{1*2}+ \frac{x}{2*3} +\frac{x}{3*4}+... \frac{x}{99*100}= \frac{99}{50}<=> \\ <=>x( \frac{1}{1*2}+ \frac{1}{2*3}+ \frac{1}{3*4}+...+ \frac{1}{99*100})= \frac{99}{50} <=> \\ <=>x(1- \frac{1}{2}+ \frac{1}{2}- \frac{1}{3}+ \frac{1}{3}- \frac{1}{4}+...+ \frac{1}{99}- \frac{1}{100})= \frac{99}{50}<=> \\ <=>x* \frac{99}{100}= \frac{99}{50}=> \\ \\ =>x=2.$

Answer 2

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