Question

Calcule a seguinte soma S = 1 + 3/2 + 5/4 + 7/8 + ...

Answer 1

Resposta:

S = 1 + 3/2 + 5/4 + 7/8 + ...  

***3/2=1/2+1/2+1/2

***5/4=1/4+1/4+1/4+1/4+1/4

***7/8=1/8+1/8+1/8+1/8+1/8+1/8+1/8

S =1 +(1/2+1/2+1/2)+(1/4+1/4+1/4+1/4+1/4) + (1/8+1/8+1/8+1/8+1/8+1/8+1/8)+..

Arrumando

S=(1+1/2+1/4+1/8+...)+(1/2+1/4+....)+(1/2+1/4+...)+(1/4+1/8+....)+

(1/4+1/8+....)+......

Usando Sn=a1/(1-q)

S=1/(1-1/2) +(1/2)/(1-1/2) + (1/2)/(1-1/2)+....

S= 2  + 1 +  1  + 1/2 +1/2+....

S=2+ 2 +1+1/2+1/8+....

S=5 +(1/2)/(1-1/2)

S=5+1 =6

Answer 2

Resposta: S = 6

Explicação passo-a-passo:

A soma S é dada por:

S = 1 + 3/2 + 5/4 + 7/8 + 9/16 + 11/32 + 13/64 + ... + ...

Reescrevendo as parcelas de S, obteremos:

1 = 2 - 1

3/2 = 5/2 - 2/2

5/4 = 7/4 - 2/4

7/8 = 9/8 - 2/8

9/16 = 11/16 - 2/16

...

...

...

Logo:

S = 1 + 3/2 + 5/4 + 7/8 + 9/16 + ... + ... + ... =>

S = (2 - 1) + 0 + (5/2 - 2/2) + (7/4 - 2/4) + (9/8 - 2/8) + (11/16 - 2/16) + ... + ... + ... =>

S = 2 - 1 + 3 - 3 + (5/2 + 7/4 + 9/8 + 11/16 + ... + ... + ...) - 2/2 - 2/4 - 2/8 - 2/16 - ... - ... - ... =>

S = (2 + 3 + 5/2 + 7/4 + 9/8 + 11/16 + ... + ... + ...) - 1 - 3 - 2(1/2 + 1/4 + 1/8 + 1/16 + ... + ... + ...) =>

S = 2(1 + 3/2 + 5/4 + 7/8 + 9/16 + 11/32 + ... + ... + ...) - 4 - 2(1/2 + 1/4 + 1/8 + 1/16 + ... + ... + ...) =>

S = 2S - 4 - 2[(1/2)/(2/2 - 1/2)] =>

S = 2S - 4 - 2[(1/2)/(1/2)] =>

S = 2S - 4 - 2 =>

S = 2S - 6 =>

S = 6

Abraços!