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

The terms of a geometric sequence can in general be written as \(a\times r^n\), where this is the \(n\)th term, and \(a\) and \(r\) are constants.

By using the information in your question you can construct two equations that you can solve to find \(a\) and \(r\).

Answer 2

I understand the formula is
\[an=a1∗r ^{n-1}\]
but when I sub in the terms given, I end up trying to do
\[\sqrt{-64} = r\]

Answer 3

The second term is -12 that means that
\[ a\times r^2=-12 \]

Answer 4

\[a * r ^{2} = -12\]\[a * r ^{5} = 768\]

Answer 5

Yes, now you can for example divide the second equation by the first, this means that you divide the left hand side of the second with the first and the same for the right hand side.

Answer 6

This is so we get rid of the \(a\), then you can solve for \(r\) directly.

Answer 7

So when dividing your equations we get
\[\frac{a*r^5}{a*r^2}=\frac{768}{-12} \]
see if you can find \(r\)

Answer 8

\[r ^{3} = -64\] but then you can't have a root of a negative number...

Answer 9

You can't have the square-root of a negative number, but this is not a square-root :p

Answer 10

For cube-roots you can move the minus sign outside the root.

Answer 11

oh, okay. That helps a lot XP \[64 = 4*4*4\]\[r = 4\]

Answer 12

You forgot the minus sign. For example, the cube-root of -8 is \(\sqrt[3]{-8}=-\sqrt[3]{8}=-2\). So if you multiply \(-2*-2*-2\) you get \(-8\).

Answer 13

sorry, no that's wrong \[r = -4\]

Answer 14

realized as soon as I hit post XP

Answer 15

Great! Now use one of your equations which you find easiest with \(-4\) in place of \(r\) to get \(a\).

Answer 16

\[a * r ^{2} = -12\]\[a * (-4)^{2} = 12\]\[a * 16 = -12\]\[a = -12 / 16\]\[a = -0.75\]

Answer 17

That's right, good job :)
This method will always work when you have two terms of a geometric sequence.

Answer 18

thank you!