Răspuns :

Madalin01vaicarid

Răspuns:

1

Explicație pas cu pas:

S=1+2+2²+...+2⁹⁹

S=2⁰+2¹+2²+....+2⁹⁹/*2 (inmultim totul cu 2)

2S=2*2⁰+2*2¹+2*2²+...+2*2⁹⁹

2S=2¹⁺⁰+2¹⁺¹+2¹⁺²+...+2¹⁺⁹⁹

2S=2¹+2²+2³+...+2¹⁰⁰

Rescriem relatiile

2S=2¹+2²+2³+...+2¹⁰⁰

S=2⁰+2¹+2²+...+2⁹⁹

------------------------------ Scadem

2S-S=2¹-2⁰+2²-2¹+2³-2²+...+2¹⁰⁰-2⁹⁹

Observam ca termenii se reduc unul cate unul, cu exceptia termenilor unici, adica a lui -2⁰ si +2¹⁰⁰

=> S=2¹⁰⁰-2⁰=2¹⁰⁰-1

=> (1+2+2²+...+2⁹⁹):(2¹⁰⁰-1)=(2¹⁰⁰-1):(2¹⁰⁰-1)=1

 

[tex]\displaystyle\bf\\(1+2+2^2+.....+2^{99}) : (2^{100}-1)=\\\\=\frac{1+2+2^2+.....+2^{99}}{2^{100}-1}=\frac{\dfrac{2^{99+1}-1}{2-1} }{2^{100}-1}=\frac{2^{99+1}-1}{2^{100}-1}=\frac{2^{100}-1}{2^{100}-1}=\boxed{\bf1}[/tex]

 

 

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