Question

Fie mulţimile A = (x € N|x = a²a€N si a < sau egal cu 5)
B={y€N|y=5n +7,n € N si n <sau egal cu 4) și C=(z€N|8 la z×2 < sau egal cu 2²⁰²³ }
a) Calculati A intersectat B.
b) Determinaţi suma elementelor mulţimii C.​

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

a)

$A = \Big\{x \in \mathbb{N} \Big| x = a^{2} \ , a \in \mathbb{N}\ , a \leq 5\Big\}$

$a \in \{0;1;2;3;4;5\}$

$0^{2}=0; \ 1^{2}=1; \ 2^{2}=4; \ 3^{2}=9; \ 4^{2}=16; \ 5^{2}=25$

$\implies A = \Big\{0;1;4;9;16;25\Big\}$

$B = \Big\{y \in \mathbb{N} \Big| y = 5n + 7, \ n \in \mathbb{N} \ , n \leq 4\Big\}$

$n \in \{0;1;2;3;4\}$

$y = 5 \cdot 0 + 7 = 7; \ y = 5 \cdot 1 + 7 = 12; \ y = 5 \cdot 2 + 7 = 17; \ y = 5 \cdot 3 + 7 = 22; \ y = 5 \cdot 4 + 7 = 27$

$\implies B = \Big\{7;12;17;22;27\Big\}$

$A \cap B = \Big\{0;1;4;9;16;25\Big\} \cap \Big\{7;12;17;22;27\Big\} = \O$

b)

$C = \Big\{z \in \mathbb{N}\Big| 8^{z} \cdot 2 \leq 2^{2023} \Big\}$

$8^{z} \cdot 2 \leq 2^{2023} \iff (2^{3} )^{z} \cdot 2 \leq 2^{2023}$

$2^{3z+1} \leq 2^{2023} \iff 3z+1 \leq 2023$

$3z \leq 2023 - 1 \iff 3z \leq 2022$

$z \leq 2022:3 \implies z \leq 674$

$\implies C = \Big\{0;1;2;...;674\Big\}$

$S = 0 + 1 + 2 + ... + 674 = \dfrac{674 \cdot 675}{2} =\bf 227475$