Find the general term of the geometric sequences defined as follows.
4, \dfrac{-8}{3},\dfrac{16}{9},\dfrac{-32}{27},\dfrac{64}{81}, …
The explicit formula for a geometric sequence is:
u_{n}=u_{1}\cdot \left(q\right)^{n-1}
where u_{1} is the first term of the sequence and q is the common ratio.
Calculate the common ratio:
q=\dfrac{u_{2}}{u_{1}}=\dfrac{-\dfrac{8}{3}}{4}=-\dfrac{2}{3}
The explicit formula for the given sequence is:
u_{n}=4\cdot \left( -\dfrac{2}{3} \right)^{n-1}
2, 6, 18, 54, 162, ...
The explicit formula for a geometric sequence is:
u_{n}=u_{1}\cdot \left(q\right)^{n-1}
where u_{1} is the first term of the sequence and q is the common ratio.
Calculate the common ratio:
q=\dfrac{u_{2}}{u_{1}}=\dfrac{6}{2}=3
The explicit formula for the given sequence is:
u_{n}=2\cdot \left( 3 \right)^{n-1}
6, -\dfrac{9}{2}, \dfrac{27}{8}, -\dfrac{81}{32},...
The explicit formula for a geometric sequence is:
u_{n}=u_{1}\cdot \left(q\right)^{n-1}
where u_{1} is the first term of the sequence and q is the common ratio.
Calculate the common ratio:
q=\dfrac{u_{2}}{u_{1}}=\dfrac{-\dfrac{9}{2}}{6}=-\dfrac{3}{4}
The explicit formula for the given sequence is:
u_{n}=6\cdot \left( -\dfrac{3}{4} \right)^{n-1}
\dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}, \dfrac{1}{16},\dfrac{1}{32}...
The explicit formula for a geometric sequence is:
u_{n}=u_{1}\cdot \left(q\right)^{n-1}
Where u_{1} is the first term of the sequence and q is the common ratio.
Calculate the common ratio:
q=\dfrac{u_{2}}{u_{1}}=\dfrac{\dfrac{1}{4}}{\dfrac{1}{2}}=\dfrac{1}{2}
The explicit formula for the given sequence is:
u_{n}= \left( \dfrac{1}{2} \right)^{n}
6, -12, 24, -48, 96,...
The explicit formula for a geometric sequence is:
u_{n}=u_{1}\cdot \left(q\right)^{n-1}
Where u_{1} is the first term of the sequence and q is the common ratio.
Calculate the common ratio:
q=\dfrac{u_{2}}{u_{1}}=\dfrac{-12}{6}=-2
The explicit formula for the given sequence is:
u_{n}= 6\cdot\left( -2 \right)^{n-1}
-4,-5,-\dfrac{25}{4},-\dfrac{125}{16},...
The explicit formula for a geometric sequence is:
u_{n}=u_{1}\cdot \left(q\right)^{n-1}
Where u_{1} is the first term of the sequence and q is the common ratio.
Calculate the common ratio:
q=\dfrac{u_{2}}{u_{1}}=\dfrac{5}{4}
The explicit formula for the given sequence is:
u_{n}= -4\cdot\left( \dfrac{5}{4} \right)^{n-1}
27, 18, 12, 8, \dfrac{16}{3},...
The explicit formula for a geometric sequence is:
u_{n}=u_{1}\cdot \left(q\right)^{n-1}
Where u_{1} is the first term of the sequence and q is the common ratio.
Calculate the common ratio:
q=\dfrac{u_{2}}{u_{1}}=\dfrac{18}{27}=\dfrac{2}{3}
The explicit formula for the given sequence is:
u_{n}= 27\cdot\left( \dfrac{2}{3} \right)^{n-1}