Question

Determinati numarul ,,n" pentru care avem:
a) $2^{n} \ \textless \ 50$
b) $b) 3^{n} \ \textless \ 90$
c)$23\ \textless \ 2^{n} \ \textless \ 90$
d) $3^{n} \ \textless \ 30 \ \textless \ 30^{n+1}$

Answer 1

Presupunand ca n este numar natural avem

a)
$2^n\ \textless \ 50 \\ Observam\ ca\ \\ 2^4=16\ \textless \ 50 \\ 2^5=32\ \textless \ 50 \\ 2^6=64\ \textgreater \ 50 \\ Deci\ n\in\{0,1,2,3,4,5\}$
b)
$3^n\ \textless \ 90 \\ Observam\ ca \\ 3^2=9\ \textless \ 90 \\ 3^3=27\ \textless \ 90 \\ 3^4=81\ \textless \ 90 \\ 3^5=81\cdot3\ \textgreater \ 90 \\ Deci\ n\in\{0,1,2,3,4\}$
c)
$23\ \textless \ 2^n\ \textless \ 90 \\ Observam\ ca \\ 2^4=16\ \textless \ 23 \\ 2^5=32;\ 23\ \textless \ 32\ \textless \ 90 \\ 2^6=64;\ 23\ \textless \ 64\ \textless \ 90 \\ 2^7=128\ \textgreater \ 90 \\ Deci\ n\in\{5,6\}$
d)
$3^n\ \textless \ 30\ \textless \ 30^{n+1} \\ Observam\ ca\ pentr\ n=0\ avem \\ 3^0\ \textless \ 30\ \textless \ 30^{0+1} \\ 1\ \textless \ 30\ \textless \ 30\ Fals\ pentru\ ca\ 30=30. \\ Pentru\ n=1\ avem \\ 3^1\ \textless \ 30\ \textless \ 30^2\ Adevarat \\ Pentru\ n=2\ avem \\ 3^2\ \textless \ 30\ \textless \ 30^3 \\ 9\ \textless \ 30\ \textless \ 30^3\ Adevarat \\ Pentru\ n=3\ avem \\ 3^3\ \textless \ 30\ \textless \ 30^4 \\ 27\ \textless \ 30\ \textless \ 30^4\ Adevarat \\ Pentru\ n=4\ avem \\ 3^4\ \textless \ 30\ \textless \ 30^5 \\ 81\ \textless \ 30\ \textless \ 30^5\ Fals\ pentru\ ca\ 81\ \textgreater \ 30. \\ Deci\ n\in\{1,2,3\}$