Question

Chứng minh các đẳng thức sau:
a) (2 + 1) (2² + 1) (2⁴ + 1) (2⁸ + 1) (2¹⁶ + 1) = 2³² - 1

b) 100² + 103² + 105² + 94² = 101² + 98² + 96² + 107²

Answer 1

\(\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

\(=\left(2^8-1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

\(=\left(2^{16}-1\right)\left(2^{16}+1\right)\)

\(=2^{32}-1\)

b/ \(100^2+\left(100+3\right)^2+\left(100+5\right)^2+\left(100-6\right)^2\)

\(=100^2+100^2+100^2+100^2+4.100+9+25+36\)

\(=100^2+2.100+1+100^2-4.100+4+100^2-8.100+16+100^2+14.100+49\)

\(=\left(100+1\right)^2+\left(100-2\right)^2+\left(100-4\right)^2+\left(100+7\right)^2\)