Question

Determinați numărul natural n pentru care:
a)3^n< 129
b)14<2^n<129
c)3^n+1<57<3^n+2
d)7^n-2≤343​

Answer 1

Explicație pas cu pas:

a)

${3}^{n} < 129 \\ {3}^{4} = 81 < 129 < 243 = {3}^{5} \\ {3}^{n} \leqslant {3}^{4} \implies \bf n \in \Big\{0 ;1 ;2 ;3 ;4\Big\}$

b)

$14 < {2}^{n} < 129 \\ {2}^{3} = 8 < 14 < 16 = {2}^{4} \\ {2}^{7} = 128 < 129 < 256 = {2}^{8} \\ {2}^{4} \leqslant {2}^{n} \leqslant {2}^{7} \implies\bf n\in \Big\{4; 5; 6; 7\Big\}$

c)

${3}^{n + 1} < 57 < {3}^{n + 2} \\ {3}^{3} = 27 < 57 < 81 = {3}^{4} \\ {3}^{n + 1} < 57 \implies (n+1)\in \Big\{0 ;1 ;2 ;3\Big\} \implies n\in \Big\{0 ;1 ;2\Big\}\\ 57 < {3}^{n + 2} \implies (n+2) \in \Big\{4 ;5 ;6 ;...\Big\} \implies n \in \Big\{2;3;4;...\Big\}\\ \implies n \in \Big\{0 ;1 ;2\Big\} \cap \Big\{2;3;4;...\Big\} \\ \implies \bf n \in \Big\{2\Big\}$

d)

${7}^{n - 2} \leqslant 343 = {7}^{3} \\ n - 2 \leqslant 3 \\ n \leqslant 5\implies \bf n\in \Big\{0 ;1 ;2 ;3 ;4;5\Big\}$