a: Ta có: \(A=100^2-99^2+98^2-97^2+\cdots+2^2-1^2\)
=(100-99)(100+99)+(98-97)(98+97)+...+(2-1)(2+1)
=100+99+...+2+1
\(=100\cdot\frac{101}{2}=50\cdot101=5050\)
b: \(B=3\left(2^2+1\right)\left(2^4+1\right)\cdot\ldots\cdot\left(2^{64}+1\right)+1\)
\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\cdot\ldots\cdot\left(2^{64}+1\right)+1\)
\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right)\cdot\ldots\cdot\left(2^{64}+1\right)+1\)
\(=\left(2^8-1\right)\left(2^8+1\right)\left(2^{16}+1\right)\cdot\left(2^{32}+1\right)\cdot\left(2^{64}+1\right)+1\)
\(=\left(2^{16}-1\right)\left(2^{16}+1\right)\cdot\left(2^{32}+1\right)\cdot\left(2^{64}+1\right)+1\)
\(=\left(2^{32}-1\right)\cdot\left(2^{32}+1\right)\cdot\left(2^{64}+1\right)+1\)
\(=\left(2^{64}-1\right)\cdot\left(2^{64}+1\right)+1=2^{128}-1+1=2^{128}\)
c: \(C=\left(a+b+c\right)^2+\left(a+b-c\right)^2-2\left(a+b\right)^2\)
\(=\left(a+b\right)^2+2c\left(a+b\right)+c^2+\left(a+b\right)^2-2c\left(a+b\right)+c^2-2\left(a+b\right)^2\)
\(=2c^2\)
