In a geometric sequence, t4=8748 and t11= -4. Determine the general termformula. Express all the answers in lowest rational form

Question

 In a geometric sequence, t4=8748 and t11= -4. Determine the general termformula. Express all the answers in lowest rational form

Answer 1

The nth term of a geometric sequence is ar^n-1

 

a = the first term

r = the common ratio between terms

 

You don't know either a or r, but you know two terms, the 4th and the 11th. 

 

It takes 7 steps to get from the 4th term to the 11th, which means that you have to multiply the 4th by common ratio 7 times to get to the 11th.  Another way to say that is:

 

t₁₁ = t₄ * r⁷

 

That let's us figure out r:

 

t₁₁ = t₄ * r⁷

-4 = 8748 r⁷

-4 / 8748 = r⁷

(-4 / 8748)^1/7 = r

 

Do this on a calculator and you get:

 

-0.33333...  = r

 

You can recognize that as -1/3.  So now we know r.

 

Let's use one of the terms we know to find a:

 

t₄ = ar^{4 - 1} = ar³

 

8748 = a (-1/3)³

8748 = a (-1/27)

-27 * 8748 = a

-236196 = a

 

So the general formula is:

 

t_n = -236196 * (-1/3)^n-1

 

You can also write this as:

 

t_n = -236196 / (-3)^n-1