Question

Dau coroana si 100 de puncte. Cine stie, va rog sa ma ajute!​

Answer 1

Răspuns:

Explicație pas cu pas:

1) 1/3(-2(2^2012 -1) +2^2013 =

1/3(-2^2013 + 2 +2^2013) = 2/3

2) 1/5(3^2013 -2(3(3^2012 -1)) =

1/5(3^2013 -2*3^2013 +6)=

1/5(-3^2013 +6)

Answer 2

Răspuns:

progresie geometrică

Explicație pas cu pas:

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

i)

$n = 2012; \ \ b_{1} = 2; \ \ q = 2$

$\frac{ {2}^{2013} }{3} - \frac{ {2}^{2012} }{3} - \frac{ {2}^{2011} }{3} - ... - \frac{2}{3} =$

$= \frac{{2}^{2013}}{3} - \frac{1}{3} \cdot ({2}^{2012} + {2}^{2011} + ... + 2)$

$= \frac{{2}^{2013}}{3} - \frac{1}{3} \cdot (2 + {2}^{2} + ... + {2}^{2011} + {2}^{2012} )$

$= \frac{{2}^{2013}}{3} - \frac{1}{3} \cdot \frac{ {2}^{2013} - 2}{1} = \frac{{2}^{2013}}{3} - \frac{{2}^{2013} - 2}{3}$

$= \frac{{2}^{2013} - {2}^{2013} + 2}{3} = \red {\bf \frac{2}{3}}$

j)

$n = 2012; \ \ b_{1} = 3; \ \ q = 3$

$\frac{ {3}^{2013} }{5} - \frac{2 \cdot {3}^{2012} }{5} - \frac{2 \cdot {3}^{2011} }{5} - ... - \frac{2 \cdot 3}{5} =$

$= \frac{{3}^{2013}}{5} - \frac{2}{5} \cdot ({3}^{2012} + {3}^{2011} + ... + 3)$

$= \frac{{3}^{2013}}{5} - \frac{2}{5} \cdot \frac{3 \cdot ( {3}^{2012} - 1) }{3 - 1}$

$= \frac{{3}^{2013}}{5} - \frac{2}{5} \cdot \frac{ {3}^{2013} - 3}{2} = \frac{{3}^{2013}}{5} - \frac{ {3}^{2013} - 3}{5}$

$= \frac{{3}^{2013} - {3}^{2013} + 3}{5} = \red{ \bf \frac{3}{5} }$