Let f (x) = x / (a (x + 2)), equation x = f (x) has unique solution, where real number a is constant, f (x1) = 2 / 2013, f (x (n)) = x (n + 1) 1. Find the expression of F (x) 2. Find the value of X (2012)

Let f (x) = x / (a (x + 2)), equation x = f (x) has unique solution, where real number a is constant, f (x1) = 2 / 2013, f (x (n)) = x (n + 1) 1. Find the expression of F (x) 2. Find the value of X (2012)

∵ f(x)=x/(a(x+2)),x=f(x)
∴ x/(a(x+2))=x
Finally, AX2 + (2a-1) x = 0;
Because there is only one solution: so Δ = (2a-1) 2 = 0;
∴ a = 0.5;
∴ f(x) = 2x/(x+2);
2 ∵x=f(x) ,F(x1)=2/2013
∴ x1 = 2/2013

2 × 1 / 3 + 3 × 1 / 4 + +2013 × 2014 1 / 2

2 × 1 / 3 + 3 × 1 / 4 + +2013 × 2014 1 / 2
(1)
=2*(1/3)+3*(1/4)+.+2013*(1/2014)
=2/3+3/4+.+2013/2014
=(1-1/3)+(1-1/4)+.+(1-1/2014)
=1-1/3+1-1/4+.+1-1/2014
=2012-1/3-1/4-.-1/2014
=2012-(1/3+1/4+.+1/2014)
=2012-6.6853419798.
=2005.3146580202.
(2)
=1/(2*3)+1/(3*4)+.+1/(2013*2014)
=(1/2-1/3)+(1/3-1/4)+.+(1/2013-1/2014)
=1/2-1/3+1/3-1/4+.+1/2013-1/2014
=1/2-1/2014
=1007/2014-1/2014
=1006/2014
=503/1007

Using simple algorithm 4.7 + 9.6 + 5.3=

=4.7+9.6+5+0.3
=4.7+0.3+5+9.6
=19.6

A simple algorithm of 5 / 7-4 / 9 + 3 / 7

5/7 + 3/7 -4/9＝8/7 - 4/9＝44/63