Question

Find an equation for the nth term of a geometric sequence where the second and fifth terms are -24 and 1536, respectively.

an = 6 • (-4)n + 1

an = 6 • 4n

an = 6 • 4n - 1

an = 6 • (-4)n - 1

Answer 1

Answer:
`a_n=6\cdot(-4)^(n-1)`

Explanation:

For geometric sequence , the nth term is given by :-

`a_n=ar^(n-1)` (1)

, where a is the first term and r is the common ratio.

As per given , we have

Second term =
`a_2=ar^(1)=-24`

Fifth term :
`a_5=ar^(4)=1536`

Divide the fifth term by Second term , we get

`(ar^(4))/(ar)=(1536)/(-24)\\\\\Rightarrow\ r^3=-64\\\\\Rightarrow\ r=\sqrt[3]{-64}= \sqrt[3]{(-4)^3}=-4`

Put value of r in second term , we get

`a(-4)=-24\\\\\Rightarrow\ a=(-24)/(-4)=6`

Now, put values of a and r in (1), we get

`a_n=6\cdot(-4)^(n-1)`

Hence, the equation for the nth term of the geometric sequence:

`a_n=6\cdot(-4)^(n-1)`