Question

Poate cineva sa-mi explice cum se fac adunarile de tipul 2¹+2²+2³+...+2¹⁰⁰ va rog?​

Answer 1

Explicație pas cu pas:

$S = {2}^{1} + {2}^{2} + {2}^{3} + ... + {2}^{100}$

$2S = 2 \cdot ({2}^{1} + {2}^{2} + {2}^{3} + ... + {2}^{100})$

$2S = {2}^{2} + {2}^{3} + {2}^{4} + ... + {2}^{100} + {2}^{101}$

$2S + 2 = \underbrace{{2}^{1} + {2}^{2} + {2}^{3} + {2}^{4} + ... + {2}^{100}}_{S} + {2}^{101}$

$2S + 2 = S + {2}^{101}$

$\red {\bf S = {2}^{101} - 2 = 2({2}^{100} - 1)}$

sau:

progresie geometrică

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

$b_{1} = 2; q = 2; n = 100$

$S = \dfrac{2 \cdot \Big({2}^{100} - 1\Big)}{2 - 1} = \dfrac{2 \cdot \Big({2}^{100} - 1\Big)}{1} =$

$= \bf 2 \cdot \Big({2}^{100} - 1\Big)$