Question

Aratati ca
1^2<1^20+1^21+1^22+...+1^39<1
REPEDE VA ROG!!!!​

Answer 1

Răspuns:

Explicație pas cu pas:

Vom demonstra ca enuntul NU este adevarat:

1 la orice putere este 1 si astfel avem

1 < 1 + 1 + ... + 1(de 20 de ori) < 1

1 < 20 < 1, ceea ce este fals pt ca 20 nu este mai mic decat 1.

Answer 2

$\it \dfrac{1}{20}+\dfrac{1}{21}+\dfrac{1}{22}+\dfrac{1}{23}+\ ...\ +\dfrac{1}{39} > \underbrace{\ \it\dfrac{1}{39}+\dfrac{1}{39}+\dfrac{1}{39}+\ ...\ \dfrac{1}{39}}_{20\ termeni}=\dfrac{20}{39} > \dfrac{20}{40}=\dfrac{1}{2}\ \ \ (1)\\ \\ \\ \dfrac{1}{20}+\dfrac{1}{21}+\dfrac{1}{22}+\dfrac{1}{23}+\ ...\ +\dfrac{1}{39} < \underbrace{\ \it\dfrac{1}{20}+\dfrac{1}{20}+\dfrac{1}{20}+\ ...\ \dfrac{1}{20}}_{20\ termeni}=\dfrac{20}{20}=1\ \ \ \ \ (2)$

$\it \\ \\ \\ (1),\ (2) \Rightarrow \dfrac{1}{2} < \dfrac{1}{20}+\dfrac{1}{21}+\dfrac{1}{22}+\dfrac{1}{23}+\ ...\ +\dfrac{1}{39} < 1$