Question

Alguien me puede ayudar?

Answer 1

Observe that

`(2 + 1) + 1 = 2 + 2 = 2^2 \implies 2 + 1 = 2^2 - 1`

`\implies (2 + 1) (2^2 + 1) = (2^2 - 1) (2^2 + 1) = 2^4 - 1`

`\implies (2 + 1) (2^2 + 1) (2^4 + 1) = (2^4 - 1) (2^4 + 1) = 2^8 - 1`

`\implies (2 + 1) (2^2 + 1) (2^4 + 1) (2^8 + 1) = (2^8 - 1) (2^8 + 1) = 2^(16) - 1`

and so on. More generally, for
`n\in\Bbb N`,

`(2 + 1) (2^2 + 1) (2^4 + 1) \cdots \left(2^(2^n) + 1\right) = 2^{2^(n+1)} - 1`

It follows that the given expression reduces to

`\frac{(2+1) (2^2 + 1) (2^4 + 1) \cdots \left(2^{2^(10)} + 1\right) + 1}{2^(26)} = \frac{2^{2^(11)}}{2^(26)} = (2^(2048))/(2^(26)) = \boxed{2^(2022)}`