Oblicz brakujące wyrazy ciągu geometrycznego. a)…,…,-27,-9,…,… b) 8, 2, …,…,… ,…, 1/512 c) …,…, 3, …,… , -81 d)…,…,…, 18,…,54 e)…, 6,… ,…, 48,… f)…,…,…,…,3,5.
źródło:
a)
a_{n+1}=a_n*q
q=\frac{a_{n+1}}{a_n}=\frac{-9}{-27}=\frac{1}{3}
a_2=\frac{-27}{\frac{1}{3}}=-81
a_1=\frac{-81}{\frac{1}{3}}=-243
a_5=-9*\frac{1}{3}=-3
a_6=-3*\frac{1}{3}=-1
-243;-81;-27;-9;-3;-1
b)
q=\frac{2}{8}=\frac{1}{4}
a_3=2*\frac{1}{4}=\frac{1}{2}
a_4=\frac{1}{2}*\frac{1}{4}=\frac{1}{8}
a_5=\frac{1}{8}*\frac{1}{4}=\frac{1}{32}
a_6=\frac{1}{32}*\frac{1}{4}=\frac{1}{128}
a_7=\frac{1}{128}*\frac{1}{4}=\frac{1}{512}
8;2;\frac{1}{2};\frac{1}{8};\frac{1}{32};\frac{1}{128};\frac{1}{512}
a_n=3
a_{n+3}=-81
a_{n+3}=a_n*q^{n+3-n}=a_n*q^3
q^3=\frac{-81}{3}=-27
q=\sqrt[3]{-27}= -3
a_2=\frac{3}{-3}=-1
a_1=\frac{-1}{-3}=\frac{1}{3}
a_4=3* (-3)= -9
a_5=-9*(-3)=27
a_6=27*(-3)=-81
\frac{1}{3};-1;3;-9;27;-81
d)
q^2=\frac{54}{18}=3
q=\sqrt3
a_5=18\sqrt3
a_6=18\sqrt3*\sqrt3=18*3=54
a_3=\frac{18}{\sqrt3}* \frac{\sqrt3}{\sqrt3}=6\sqrt3
a_2=\frac{6\sqrt3}{\sqrt3}=6
a_1=\frac{6}{\sqrt3}*\frac{\sqrt3}{\sqrt3}=2\sqrt3
2\sqrt3;6;6\sqrt3;18;18\sqrt354
e)
q^3=\frac{48}{6}=8
q=\sqrt[3]8=2
a_1=\frac{6}{2}=3
a_3=6*2=12
a_4=12*2=24
a_5=24*2=48
a_6=48*2=96
3;6;12;24;48;96
f)
q=\frac{5}{3}
a_4=\frac{3}{\frac{5}{3}}=3* \frac{3}{5}=\frac{9}{5}
a_3=\frac{\frac{9}{5}}{\frac{5}{3}}=\frac{9}{5}*\frac{3}{5}=\frac{27}{25}
a_2=\frac{\frac{27}{25}}{\frac{5}{3}}=\frac{27}{25}*\frac{3}{5}=\frac{81}{125}
a_1=\frac{\frac{81}{125}}{\frac{5}{3}}=\frac{81}{125}*\frac{3}{5}=\frac{243}{625}
\frac{243}{625};\frac{81}{125};\frac{27}{25};\frac{9}{5};3;5