Liczba 20, x, 40 sa trzema poczatkowymi wyrazami ciagu geometrycznego o wyrazach dodatnich. Oblicz x i sume szesnastu poczatkowych wyrazow tego ciagu.

Odpowiedź :

[tex]\left \{ {{20q = x} \atop {qx = 40}} \right. \\\left \{ {{20q = x} \atop {q*20q = 40}} \right. \\\left \{ {{20q = x} \atop {20q^{2} = 40}} \right. \\\left \{ {{20q = x} \atop {q^{2} = 2}} \right. \\\left \{ {{20q = x} \atop {q = \sqrt{2} }} \right. \\\left \{ {{x = 20\sqrt{2} } \atop {q = \sqrt{2} }} \right.[/tex]

[tex]S_{16} = 20 * \frac{1-\sqrt{2}^{16} }{1-\sqrt{2} } = 20 * \frac{1-256}{1-\sqrt{2} } = \frac{20*(-255)}{1-\sqrt{2} } = \frac{-5100}{1-\sqrt{2} } = \frac{-5100 - 5100\sqrt{2} }{1-2} = \frac{-5100-5100\sqrt{2} }{-1} = 5100 + 5100\sqrt{2} = 5100(1 + \sqrt{2})[/tex]