Ciągi geometryczne

[matmabee]

Oblicz q i n
a) a\(_1=729\), \(a_n=96\), \(S_n=1995\)
b) \(a_1=128\), \(a_n=1458,\) \(S_n=4118\)

[eresh]

Oblicz q i n
a) a_1=729, a_n=96, S_n=1995

\(a_1q^{n-1}=a_n\\
729\cdot q^{n-1}=96\\
q^{n-1}=\frac{32}{243}\)

\(\frac{a_1(1-q^n)}{1-q}=1995\\
\frac{729(1-q^n)}{1-q}=1995\\
1-q^n=\frac{665}{243}(1-q)\\
1-q^n=\frac{665}{243}-\frac{665}{243}q\\
q^n=-\frac{422}{243}+\frac{665}{243}q\)

\(a_1q^{n-1}=a_n\\
729\cdot q^{n-1}=96\\
q^{n-1}=\frac{32}{243}\\
q^n\cdot q^{-1}=\frac{32}{243}\\
\frac{-\frac{422}{243}+\frac{665}{243}q}{q}=\frac{32}{243}\\
-\frac{422}{243}+\frac{665}{243}q=\frac{32}{243}q\\
-\frac{422}{243}=-\frac{633}{243}q\\
q=\frac{2}{3}\\

\)

\(q^{n-1}=\frac{32}{243}\\
(\frac{2}{3})^n\cdot\frac{3}{2}=\frac{32}{243}\\
(\frac{2}{3})^n=\frac{64}{729}\\
n=6\)

[eresh]

Oblicz q i n

b) a_1=128,a_n=1458, S_n=4118

\(\frac{a_1(1-q^n)}{1-q}=4118\\
\frac{128(1-q^n)}{1-q}=4118\\
1-q^n=\frac{4118}{128}(1-q)\\
1-q^n=\frac{4118}{128}-\frac{4118}{128}q\\
q^n=-\frac{1995}{64}+\frac{2059}{64}q\)

\(a_1q^{n-1}=a_n\\
128\cdot q^{n-1}=1458\\
q^{n-1}=\frac{729}{64}\\
q^n\cdot q^{-1}=\frac{729}{64}\\
\frac{-\frac{1995}{64}+\frac{2059}{64}q}{q}=\frac{729}{64}\\
-\frac{1995}{64}+\frac{2059}{64}q=\frac{729}{64}q\\
\frac{1330}{64}=\frac{1995}{64}\\
q=\frac{3}{2}\\

\)

\(128\cdot q^{n-1}=1458\\
(\frac{3}{2})^{n-1}=\frac{729}{64}\\
n-1=6\\
n=7\)