Question

find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -12 and 768 respectively

Answer 1

`a_n=3\cdot(-4)^(n-1)`

Explanation:

We have the following formula for nth terms

`a_n=a_1\cdot r^(n-1)^{}`

we replace for each point and we are left

`\begin{gathered} a_2=-12 \\ -12=a_1\cdot r^(2-1)\rightarrow-12=a_1\cdot r^{}\text{ (1)} \\ a_5=768 \\ 768=a_1\cdot r^(5-1)\rightarrow768=a_1\cdot r^4\text{ (2)} \end{gathered}`

We solve the system of equations that remains like this:

`\begin{gathered} a_1=(-12)/(r)\text{ (3)} \\ a_1=(768)/(r^3)\text{ (4)} \\ \text{we equalize (3) and (4)} \\ -(12)/(r)=(768)/(r^4) \\ r^3=(768)/(-12) \\ r=\sqrt[3]{-64} \\ r=-4 \end{gathered}`

Now, for a1

`\begin{gathered} a_1=(-12)/(-4) \\ a_1=3 \end{gathered}`