Question

Exercițiul 21 .......​

Answer 1

(...(((2+2)×2+2³)×2+2⁵)×2+...+2²⁰⁰¹)×2+2²⁰⁰³

= (...((4×2+2³)×2+2⁵)×2+...+2²⁰⁰¹)×2+2²⁰⁰³

= (...((8+2³)×2+2⁵)×2+...+2²⁰⁰¹)×2+2²⁰⁰³

= (...(16×2+2⁵)×2+...+2²⁰⁰¹)×2+2²⁰⁰³

= (...(64×2+2⁷)×2+...+2²⁰⁰¹)×2+2²⁰⁰³

= (...(256×2+2⁹)×2+...+2²⁰⁰¹)×2+2²⁰⁰³

= (...(1024×2+2¹¹)×2+...+2²⁰⁰¹)×2+2²⁰⁰³

Remarca:

16×2+2⁵=2⁴×2¹+2⁵=2⁵+2⁵=2⁶

64×2+2⁷=2⁶×2¹+2⁷=2⁷+2⁷=2⁸

256×2+2⁹=2⁸×2¹+2⁹=2⁹+2⁹=2¹⁰

1024×2+2¹¹=2¹⁰×2¹+2¹¹=2¹¹+2¹¹=2¹²

$= > {2}^{x} \times 2 + {2}^{x + 1} = {2}^{x + 2}$

Ultimul element din paranteza: 2²⁰⁰¹ = 2^x+1 => 2^x = 2²⁰⁰⁰; 2^x+2 = 2²⁰⁰²

=> (2²⁰⁰⁰×2+2²⁰⁰¹)×2+2²⁰⁰³=2²⁰⁰²×2+2²⁰⁰³=2²⁰⁰⁴