Question

Aratati ca nr. urmatore sunt patrate perfecte:
c) 2^1999-2^1998-2^1997-2^1996=
d) 1+3+5+....2005=

Answer 1

$c)\quad 2^{1999}-2^{1998}-2^{1997}-2^{1996} = \\ \\ =2^{1996}\times(2^3-2^2-2^1-2^0) = \\ \\ = 2^{1996}\times(8-4-2-1) = \\ \\ = 2^{1996}\times1 = \\ \\ = 2^{1996} = \\ \\ ={\big(2^{998}\big)}^2 \rightarrow p.p.$


$d)\quad \boxed{1+3+5+...+(2\cdot n-1)=n^2} \rightarrow formula\\ \\1+3+5+...+2005 = \\ \\ = 1+3+5+...+\big(2\cdot1003-1\big) = 1003^2\rightarrow p.p.$