Question

calculați S indice 1 = 1+1/2 + 1/2^2+..... +1/2^n-1​

Answer 1

Răspuns:

$s = \frac{ {2}^{n} - 1 }{ {2}^{n - 1} }$

Explicație pas cu pas:

$s = 1 + \frac{1}{2} + \frac{1}{ {2}^{2} } + .... + \frac{1}{ {2}^{n - 1} }$

$\frac{1}{2} s = \frac{1}{2} + \frac{1}{ {2}^{2} } + \frac{1}{ {2}^{3} } + ..... + \frac{1}{ {2}^{n} }$

Le scădem:

$\frac{1}{2} s = 1 - \frac{1}{ {2}^{n} }$

$\frac{1}{2} s = \frac{ {2}^{n} - 1}{ {2}^{n} }$

$s = \frac{ \frac{ {2}^{n} - 1}{ {2}^{n} } }{ \frac{1}{2} }$

$s = 2 \times \frac{ {2}^{n} - 1}{ {2}^{n} }$

$s = \frac{ {2}^{n} - 1}{ {2}^{n - 1} }$