geometric sequences

[1141]

I'm lost on how to do this:

Find the common ratio and the first term in the geometric progressions where:

a.) the 2nd term is 4 and the 5th term is 108
b.) the 3rd term is 8 and the 9th term is 64

can anyone help?

 

[soroban]

\(\displaystyle \text{Hello, 1141!}\)

\(\displaystyle \text{You're expected to know the formula for the }n^{th}\text{ term: }\;\;a_n \:=\:ar^{n-1}\)
. . \(\displaystyle \text{where: }\,a\text{ = first term, }\,r\text{ = common ratio.}\)

Find the common ratio and the first term in the geometric progressions where:

a) the 2nd term is 4 and the 5th term is 108

\(\displaystyle \text{We are given: }\;\begin{array}{cccccccccc}a_2 &=& 4 && \Rightarrow && ar &=& 4 & [1] \\ a_5 &=& 108 && \Rightarrow && ar^4 &=& 108 & [2] \end{array}\)


\(\displaystyle \text{Divide [2] by [1]: }\;\frac{ar^4}{ar} \;=\;\frac{108}{4} \quad\Rightarrow\quad r^3 \:=\:27 \quad\Rightarrow\quad\boxed{ r = 3}\)


\(\displaystyle \text{Substitute into [1]: }\;a(3) \:=\:4 \quad\Rightarrow\quad \boxed{a \:=\:\frac{4}{3}}\)