Sequences - Need help for exam question!

Question

The following terms are given:

U₁ = 196830  U₄ = 7290 and U₇ = 270

 

The starting term is U₀

 

Is this an arithmetic or geometric sequence? Motivate your answer.

 

I tried finding the common difference or common ratio to see wether it was arithmetic or geometric, but was unable to fully understand it.

 

How would one go about answering this and finding r or d for this?

Answer 1

Let's compute the "common difference" (d) between U4 and U1 and U7 and U4:

 

d = (U4-U1)/(4-1) = (7290-196830)/3 = -63,180

d = (U7-U4)/(7-4) = (270-7290)/3 = -2340

 

So it's not an arithmetic sequence because it doesn't have a common difference (d).  Let's see if the terms have a common ratio, r:

 

r = (U4/U1)^1/(4-1) = (7290/196830)^1/3 = 1/3

r = (U7-U4)^1(7-4) = (270/7290)^1/3 = 1/3

 

Last, let's check the common ratio between U7 and U1:

 

r = (u7-U1)^1(7-1) = (270/196830)^1/6 = 1/3

 

Since there is a common ratio, r=1/3, between all of the given terms, the sequence is geometric.

 

 

Answer 2

this is a geometric sequence

 

divide U₄ by U₁ and you get 1/27.

 

You get the same thing when you divide U₇ by U₄

 

this is r³ because you would have to multiply U₁ by r to get U₂, the multiply by r to get U₃ then by r again to get U₄

 

if r³ = 1/27  then r = 1/3

 

If you had U₁ and U_5, you would divide and set it equal to r⁴ because there are 4 'steps' betwenn the two terms