Question

Calculati suma:
$2^{1} + 2^2+2^3+...+2^{2014}=$

Answer 1

$S=2^1+2^2+...+2^{2014}$


Inmultim cu 2

$2S=2^2+2^3+2^4+...+2^{2015}$

Calculam $2S-S=2^2+2^3+2^4+....+2^{2015}-2^1-2^2-2^3-...-2^{2014}$

$S=2^{2015}-2=2(2^{2014}-1)$

Answer 2

Suma este o progresie geometrica.
Mai intai calculam ratia:  q=2²:2=2    => q=2
Apoi suma dupa formula:
S= b_{1} * \frac{q^n-1}{q-1} 

S=2* \frac{2^{2014}-1 }{2-1} =
=2*(2^{2014}-1)=2^{2015} -2