Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -6 and 162, respectively.

Answer 1

Let's look at the information we have here, shall we? :D

Answer 2

Let the first term of the geometric progression be \(a\) and the common ratio be \(r\). You're given that the second term is -6 and the fifth term is 162.

Express those two as equations.

Answer 3

like -6/162?

Answer 4

No. The \(n\)th term of a geometric progression is \(a \times r^{n-1}\). The \(k\)th term of a geometric progression is \(a \times r^{k-1}\). Similarly, what's the \(2\)nd term of the geometric progression?

Answer 5

-6 is the second term

Answer 6

\[a \times r^{2-1} = -6\]Right?

Answer 7

yes

Answer 8

Similarly, how would you make an equation for the 5th term?

Answer 9

a * r ^5-1 = 162?

Answer 10

Good going! You have two equations:\[a \times r = -6\]\[a \times r^4 = 162\]What you can do is divide the two to get the value of \(r\)

Answer 11

3 right?

Answer 12

-3, yes.

Answer 13

does it matter if its negative or positive?

Answer 14

wait nvm I get why its -3...its because the first term is positive..im dumb

Answer 15

Good stuff. :)

Answer 16

Now that you know what \(r\) is, you can know what \(a\) is.

Answer 17

a is 2 right?