Question

Find an equation for the nth term of a geometric sequence where the second and fifth terms are -21 and 567, respectively.

Answer 1

A geometric sequence is a number that is related to the previous number by a certain factor, say \(r\):
$$
a_n=r\times a_{n-1}=ra_{n-1}
$$
We know that \(a_2=-21\) and \(a_{5}=567\)
$$
a_2=ra_1=-21\\
a_3=ra_2=r(-21)\\
a_4=ra_3=r\times r(-21)=r^2(-21)\\
a_5=ra_4=r\times r^2(-21)=r^3(-21)=567\\
\implies r=\sqrt[1/3]{\cfrac{567}{-21}}=3
$$
This means that \(a_n=3a_{n-1}\)

Make sense?

Answer 2

*correction
$$
r=-3\\
a_n=-3a_{n-1}
$$

Answer 3

7 • (-3)^n - 1

is that the answer

Answer 4

the difference now, is instead of adding a number each time, now you are multiplying by a number each time

Answer 5

the number b, is what you multiply by each time moving up consecutive terms

they gave you the second and fifth terms, they arent consecutive...

Answer 6

an = a0*(b)^n

need to figure out a0 , and b (2 variables)

they gave you two data points
(n,an) = (2 , -21)
(n , an) = (5 , 567)

Answer 7

those will give you two equations, you can solve those for b and a0

Answer 8

-21 = a0*(b)^2

and

567 = a0*(b)^5

Answer 9

or another way
if you dont want to solve that system,

moving from the second term to the 5th term, you are multiplying by 'b' 3 times

-21*b^3 = 567

b = -3, right off the bat

an = a0*(-3)^n

Answer 10

you can use any point, either of those two equations to get a0

Answer 11

thank you but i just want to know if my answer is right

Answer 12

7 • (-3)^n - 1

Answer 13

oh, you already know all this

Answer 14

did i get it right

Answer 15

what is that -1 at the end for

Answer 16

\[7 * (-3)^{n-1}\]

Answer 17

oh yeah, sorry, i have been missing that the entire time

Answer 18

thank you!!

Answer 19

a0

-21 = ao*-3^(2-1)

a0 = 7 yeah

Answer 20

if it is too difficult to figure r, or can't, you can solve a system like above, except the pwers should be 1 less on those (n-1)