Question

1.Comparati nr :
a)$2^{42}$ cu $3^{26}$
B)5+5²+5³+....+$5^{50}$ si $2^{2975}$
2.Care dintre nr :A=$a^{1+2+...+1980} si B=( a^{990} )^{1981}$

Answer 1

a) $2^{42} \ \textless \ 3^{28}$
 $2^{3*14} \ \textless \ 3^{2*14}$
$8^{14} \ \textless \ 9^{14}$

b)[tex]5+5^2+5^3+...+5^{50} \ \textless \ 2^{2975} [/tex]

$S=5+5^2+5^3+...+5^{50} | x5$
$5S=5^2+5^3+...+5^{50}+5^{51}+5-5$
$5S=S+5^{51}-5$
$4S=5^{51}-5$
$S=(5^{51}-5):4$
deci, comparam mai departe:
$(5^{51}-5):4 \ \textless \ 2^{2975}$
$5^{51}-5 \ \textless \ 2^{2975}*2^2$
$5^{51}-5) \ \textless \ 2^{2977}$
$2^{2977}\ \textgreater \ 2^{2976}$
$5^{51} \ \textless \ 4^{1488}$
$5^{51} \ \textless \ 8^{992}$

c)$A= a^{1+2+...+1980}~si ~B=( a^{990} )^{1981}$
1+2+...+1980=1980*1981:2=990*1981
=>$A=( a^{990} )^{1981}$
=> A=B