Question

Rút gọn phân số A = \(\dfrac{25^{28}+25^{24}+25^{20}+...+25^4+1}{25^{30}+25^{28}+....+25^2+1}\)

Answer 1

Lời giải:

Xét tử số:

$\text{TS}=1+25^4+25^8+...+25^{28}$

$25^4.\text{TS}=25^4+25^8+...+25^{32}$

$\Rightarrow \text{TS}(25^4-1)=25^{32}-1$

$\text{TS}=\frac{25^{32}-1}{25^4-1}$

Xét mẫu số:

$\text{MS}=1+25^2+..+25^{30}$

$25^2.\text{MS}=25^2+25^4+...+25^{32}$

$\Rightarrow \text{MS}(25^2-1)=25^{32}-1$

$\Rightarrow \text{MS}=\frac{25^{32}-1}{25^2-1}$

Do đó:
$A=\frac{25^{32}-1}{25^4-1}:\frac{25^{32}-1}{25^2-1}=\frac{25^2-1}{25^4-1}$

$=\frac{25^2-1}{(25^2-1)(25^2+1)}=\frac{1}{25^2+1}$