Question

[Đặng Nguyên Nhật]

Cho b^2=ác. CM:a^2022+b^2022/b^2022+c^2022=(a+b/b+c)^2022

Answer 1

[Akai Haruma]

Lời giải:

$b^2=ac\Rightarrow \frac{b}{a}=\frac{c}{b}$

Đặt $\frac{b}{a}=\frac{c}{b}=k\Rightarrow b=ak; c=bk$

Khi đó:
$\frac{a^{2022}+b^{2022}}{b^{2022}+c^{2022}}=\frac{a^{2022}+(ak)^{2022}}{b^{2022}+(bk)^{2022}}$

$=\frac{a^{2022}(1+k^{2022})}{b^{2022}(1+k^{2022})}=\frac{a^{2022}}{b^{2022}} (1)$

Và:

$(\frac{a+b}{b+c})^{2022}=(\frac{a+ak}{b+bk})^{2022}$

$=[\frac{a(k+1)}{b(1+k)}]^{2022}=(\frac{a}{b})^{2022}=\frac{a^{2022}}{b^{2022}}(2)$

Từ $(1); (2)$ ta có đpcm.