Question

sa se arate ca : 1 supra 5n+1 + 1supra5n+2 +...+1 supra 10n >3 supra 5. Am nevoie urgent

Answer 1

Notez cu S=1/(5*n+1) +1/(5*n+2) +...+1/(5*n+5*n) =
=1/(5*n+1) +1/(5*n+2) +...+1/(5*n+n) +1/(6*n+1) +...+1/(6*n+n) +...1/(9*n+1)+...+1/(9*n+n)

cum :
1/(5*n+1) >1/(5*n+n)
1/(5*n+2) >1/(5*n+n)
..............................
1/(5*n+n)>= 1/(5*n+n)

=================
1/(5*n+1) +1/(5*n+2)+...+1/(5*n+n) >1/6*n+1/6*n+...+1/6*n =n/6*n=1/6
analog pt
1/(6*n+1) +1/(6*n+2)+....+1/(6*n+n)>1/7*n+...+1/7*n =n/7*n =1/7
1/(7*n+1) +...+1/(7*n+n) >1/8
1/(8*n+1) +...+1/(8*n+n) >1/9
1/(9*n+1) +...+1/(9*n+n) >1/10
deci adunate avem:
S>1/6+1/7+1/8+1/9+1/10 , aducem la numitorul comun 5*8*9*7
=>
S>( (5*4*3*7) +(5*8*9) +(5*9*7)+(8*7*5)+(4*9*7) )/(5*8*9*7) =>
=> S> (  420 +  360   +  315 +280+  252 ) /2520 =>
=> S>1627/2520 >1512/2520 =3/5