Question

Dau coroana pentru un raspuns intreg si corect!

Answer 1

Explicație pas cu pas:

$1 + \dfrac{1}{2} + \dfrac{1}{ {2}^{2} } + ... + \dfrac{1}{ {2}^{7} } =$

$= \dfrac{{2}^{7}}{ {2}^{7} } + \dfrac{{2}^{6}}{ {2}^{7} } + \dfrac{{2}^{5}}{ {2}^{7} } + ... + \dfrac{ {2}^{1} }{ {2}^{7} } + \dfrac{{2}^{0}}{ {2}^{7} }$

$= ({2}^{0} + {2}^{1} + ... + {2}^{6} + {2}^{7}) \cdot \dfrac{1}{ {2}^{7} }$

$= \dfrac{{2}^{8} - 1}{ {2}^{7} } = \bf 2 - \dfrac{1}{ {2}^{7} }$

unde:

$\boxed {S_{n} = \dfrac{b_{1}\cdot ({q}^{n} - 1) }{q - 1}}$

$b_{1} = {2}^{0} \ ; \ q = 2 \ ; \ n = 8$

${2}^{0} + {2}^{1} + ... + {2}^{6} + {2}^{7} = \dfrac{ {2}^{0} \cdot ({2}^{8} - 1) }{2 - 1} = {2}^{8} - 1$

sau:

$1 + \dfrac{1}{2} + \dfrac{1}{ {2}^{2} } + ... + \dfrac{1}{ {2}^{6} } + \dfrac{1}{ {2}^{7} } =$

$= 1 + \Big(1 - \dfrac{1}{2}\Big) + \Big(\dfrac{1}{2} - \dfrac{1}{ {2}^{2} }\Big) + ... + \Big(\dfrac{1}{ {2}^{5} } - \dfrac{1}{ {2}^{6} }\Big) + \Big(\dfrac{1}{ {2}^{6} } - \dfrac{1}{ {2}^{7} }\Big)$

$= 1 + 1 - \dfrac{1}{ {2}^{7}} = \bf 2 - \dfrac{1}{ {2}^{7}}$

Answer 2

Unde nu înțelegi mă întrebi.