Question

How do I do this? can I get a step by step so I understand well.

Answer 1

general geometric formula is

`A_n=A_1\cdot r^(n-1)`

then we replace using A3

`\begin{gathered} A_3=A_1\cdot r^(3-1) \\ \\ (16)/(3)=A_1\cdot r^2 \end{gathered}`

now replace using A5

`\begin{gathered} A_5=A_1\cdot r^(5-1) \\ \\ (64)/(21)=A_1\cdot r^4 \end{gathered}`

now we have two equations and two unknow

`\begin{gathered} (16)/(3)=A_1\cdot r^2 \\ \\ (64)/(21)=A_1\cdot r^4 \end{gathered}`

we can solve A1 or r from any equation and replace on the other

I will solve A1 from the first equation

`A_1=((16)/(3))/(r^2)`

and replace on the second to solve r

`\begin{gathered} (64)/(21)=(((16)/(3))/(r^2))\cdot r^4 \\ \\ (64)/(21)=(16)/(3)\cdot r^2 \\ \\ r^2=(64*3)/(21*16) \\ \\ r^2=(192)/(336)=(4)/(7) \\ \\ r=\frac{2\sqrt[]{7}}{7} \end{gathered}`

now replace r on the other equation to find A1

`\begin{gathered} (16)/(3)=A_1\cdot(\frac{2\sqrt[]{7}}{7})^2 \\ \\ (16)/(3)=A_1\cdot(4)/(7) \\ \\ A_1=(16*7)/(4*3) \\ \\ A_1=(28)/(3) \end{gathered}`

now we have the two unknows A1 and r then replace on the general geometric equation

`A_n=(28)/(3)\cdot(\frac{2\sqrt[]{7}}{7})^(n-1)`