Question

Fie numerele:
a=1+2+2^2+2^3+...+2^2013 si b=1+3+3^2+...+3^2013
Comparati numerele:(a+1)^3 si (2b+1)^2
Ajutati-ma!Va rog!

Answer 1

$a=1* \frac{ 2^{2014}-1 }{2-1} = 2^{2014} -1$
$b=1* \frac{3^{2014}-1}{3-1} = \frac{ 3^{2014}-1 }{2}$
$(a+1)^{3} =( 2^{2014}-1+1 )^{3}= 2^{6042}$
$(2b+1)^{2} =(2* \frac{ 3^{2014-1} }{2}+1 )^{2}= 3^{6042}$
$2^{6042} < 3^{6042} => (a+1)^{3} <(2b+1)^{2}$