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Đặt \(A=12.\left(5^2+1\right).\left(5^4+1\right).\left(5^8+1\right).\left(5^{16}+1\right)\)
\(\Rightarrow2A=24.\left(5^2+1\right).\left(5^4+1\right).\left(5^8+1\right).\left(5^{16}+1\right)\)
\(2A=\left(5^2-1\right).\left(5^2+1\right).\left(5^4+1\right).\left(5^8+1\right).\left(5^{16}+1\right)\)
\(2A=\left(5^4-1\right).\left(5^4+1\right).\left(5^8+1\right).\left(5^{16}+1\right)\)
\(2A=\left(5^8-1\right).\left(5^8+1\right).\left(5^{16}+1\right)\)
\(2A=\left(5^{16}-1\right).\left(5^{16}+1\right)\)
\(2A=\left(5^{16}\right)^2-1^2\)
\(2A=5^{32}-1\)
\(\Rightarrow A=\frac{5^{32}-1}{2}.\)
P = 12 ( 52 + 1 ) ( 54 + 1 ) ( 58 +1 ) ( 516 +1 )
=>2P=24 ( 52 + 1 ) ( 54 + 1 ) ( 58 +1 ) ( 516 +1 )
=( 52 - 1 ) ( 52 + 1 ) ( 54 + 1 ) ( 58 +1 ) ( 516 +1 )
=( 54 - 1 ) ( 54 + 1 ) ( 58 +1 ) ( 516 +1 )
=( 58 - 1 ) ( 58 +1 ) ( 516 +1 )
=( 516 - 1 ) ( 516 + 1 )
= 532 - 1
=>P=\(\frac{5^{32}-1}{2}\)
\(P=12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(2P=\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(2P=\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(2P=\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(2P=\left(5^{16}-1\right)\left(5^{16}+1\right)\)
\(2P=5^{32}-1\)
\(P=\frac{5^{32}-1}{2}\)
Giải:
a) \(A=\left(3^8+1\right)\left(3^4+1\right)\left(3^2+1\right)\left(3+1\right)\)
\(\Leftrightarrow2A=2\left(3^8+1\right)\left(3^4+1\right)\left(3^2+1\right)\left(3+1\right)\)
\(\Leftrightarrow2A=\left(3^8+1\right)\left(3^4+1\right)\left(3^2+1\right)\left(3+1\right)\left(3-1\right)\)
\(\Leftrightarrow2A=\left(3^8+1\right)\left(3^4+1\right)\left(3^2+1\right)\left(3^2-1\right)\)
\(\Leftrightarrow2A=\left(3^8+1\right)\left(3^4+1\right)\left(3^4-1\right)\)
\(\Leftrightarrow2A=\left(3^8+1\right)\left(3^8-1\right)\)
\(\Leftrightarrow2A=3^{16}-1\)
\(\Leftrightarrow A=\dfrac{3^{16}-1}{2}\)
Vậy ...
b) \(B=12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)...\left(5^{32}+1\right)\)
\(\Leftrightarrow2B=2.12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)...\left(5^{32}+1\right)\)
\(\Leftrightarrow2B=24\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)...\left(5^{32}+1\right)\)
\(\Leftrightarrow2B=\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)...\left(5^{32}+1\right)\)
\(\Leftrightarrow2B=\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)...\left(5^{32}+1\right)\)
...
\(\Leftrightarrow2B=\left(5^{32}-1\right)\left(5^{32}+1\right)\)
\(\Leftrightarrow2B=5^{64}-1\)
\(\Leftrightarrow B=\dfrac{5^{64}-1}{2}\)
Vậy ...
Ta có:
P=12(52+1)(54+1)(58+1)(516+1)
P=(52-1)(52+1)(54+1)(58+1)(516+1):2
P=(54-1)(54+1)(58+1)(516+1):2
P=(58-1)(58+1)(516+1):2
P=(516-1)(516+1):2
P=(532-1):2
\(P=12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(=\frac{1}{2}\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(=\frac{1}{2}\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(=\frac{1}{2}\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)
\(=\frac{1}{2}\left(5^{16}-1\right)\left(5^{16}+1\right)=\frac{5^{32}-1}{2}\)
Nhân cả 2 vế với 2 ta có:
2P=24(52+1)(54+1)(58+1)(516+1)
2P=(52_1)(52+1)(54+1)(58+1)(516+1)
2P=(54_1)(54+1)(58+1)(516+1)
2P=(58_1)(58+1)(516+1)
2P=(516_1)(516+1)
2P=532_1
P=\(\frac{5^{32}-1}{2}\)
\(2P=24\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)=\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)=\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)=\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)=\left(5^{16}-1\right)\left(5^{16}+1\right)=5^{32}-1\Rightarrow P=\frac{5^{32}-1}{2}\)
chúc bạn học tốt 
A = \(\frac{1}{2}.24\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
A = \(\frac{1}{2}\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
A = \(\frac{1}{2}\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
A = \(\frac{1}{2}\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
A = \(\frac{1}{2}\left(5^{16}-1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
A = \(\frac{1}{2}\left(5^{32}-1\right)\left(5^{32}+1\right)\)
A = \(\frac{1}{2}\left(5^{64}-1\right)\)
\(2A=24\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(=\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(=5^{64}-1\)
=> \(A=\frac{5^{64}-1}{2}\)