Question

1. Rút gọn các biểu thức sau:

a. A = 100² - 99²+ 98² - 97² + ... + 2² - 1²

b. B = 3(2² + 1) (2⁴ + 1) ... (2⁶⁴ + 1) + 1²

c. C = (a + b + c)² + (a + b - c)² - 2(a + b)²

2. Chứng minh rằng:

a. a³ + b³ = (a + b)³ - 3ab (a + b)

b. a³ + b³ + c³ - 3abc = (a + b + c) (a² + b² c² - ab - bc - ca)

Answer 1

Câu 1:

a: \(A=\left(100+99\right)\left(100-99\right)+\left(98+97\right)\left(98-97\right)+...+\left(2-1\right)\left(2+1\right)\)

\(=100+99+98+...+3+2+1\)

=5050

b: \(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\cdot...\cdot\left(2^{64}+1\right)+1\)

\(=\left(2^4-1\right)\cdot\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)+1\)

\(=\left(2^8-1\right)\left(2^8+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)+1\)

\(=\left(2^{16}-1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)+1\)

\(=\left(2^{32}-1\right)\left(2^{32}+1\right)\left(2^{64}+1\right)+1\)

\(=\left(2^{64}-1\right)\left(2^{64}+1\right)+1\)

\(=2^{128}\)

c: \(\left(a+b+c\right)^2+\left(a+b-c\right)^2-2\left(a+b\right)^2\)

\(=\left(a+b\right)^2+2c\left(a+b\right)+c^2+\left(a+b\right)^2-2c\left(a+b\right)+c^2-2\left(a+b\right)^2\)

\(=2c^2\)