Question

Comparați numerele:
a) 14 < 2 la puterea n <129
b) 3 la puterea n + 1 <57 < 3 la puterea n + 1
c) 3 la puterea n + 1 <57 < 3 la puterea n + 2
d) 7 la puterea n - 2 <_ 343

va rog urgent! ​​

Answer 1

Explicație pas cu pas:

a) 14 < 2 la puterea n <129

$14 < {2}^{n} < 129$

$8 = {2}^{3} < 14 < {2}^{4} = 16 = > 3 < n$

$128 = {2}^{7} < 129 < {2}^{8} = 256 = > n < 8$

$3 < n < 8 = > n \in \{4; 5; 6; 7\}$

b) 3 la puterea n + 1 <57 < 3 la puterea n + 1

${3}^{n + 1} < 57 < {3}^{n + 1}$

n ∈ Ø

c) 3 la puterea n + 1 <57 < 3 la puterea n + 2

${3}^{n + 1} < 57 < {3}^{n + 2}$

$27 = {3}^{3} < 57 < {3}^{4} = 81 \\ n + 1 < 4 = > n < 3 \\ n + 2 > 3 = > n > 1 \\ 1 < n < 3 = > n = 2$

d) 7 la puterea n - 2 <_ 343

${7}^{n - 2} \leqslant 343$

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