Question

(2+1)(\(2^2\)+1)(\(2^4\)+1)...(\(2^{20}\)+1)+1

Answer 1

\(\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)....\left(2^{32}+1\right)+1\)

\(=\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)....\left(2^{32}+1\right)+1\)

\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)...\left(2^{32}+1\right)+1\)

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

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

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

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

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

\(=2^{64}\)