Determine if the following sequence is geometric, arithmetic, neither, or not even a sequence.

Question

1,4,9,16,25,36,...

Answer 1

Taylor,

 

If you happened to notice that the sequence is s(i)=i^2 for i=1,2,3....then 

you can find an explicit formula for the ratio and differences of successive terms.

 

So the (i + 1)th term, s(i+1) is (i+1)^2 and the ith is i^2

 

The ratio is (i+1)^2/i^2=(1+1/i)^2.  This depends on i so is not constant...the requirement for geometric progression.

Therefore the sequence is not in GP.

 

The difference is (i+1)^2-i^2=2i+1 which also depends on i.
Therefore the sequence is not in AP.

 

The answer is neither.  Since we have found a rule defining the terms, s is a sequence.

 

What if we did not notice that s(i)=i^2?

 

We could still test successive terms for constant ratio or difference.  For example, 4/1=4 not equal to 9/4 and 

4-1=3 not equal to 9-3.  So on that basis the sequence fails to be in GP or AP respectively.

 

 

 

Answer 2

Hey!

 

Those numbers look very familiar to me (and they should to you, too).  They are n^2 for n=1,2,3,4,5,6.

 

So, is the sequence arithmetic (constant difference)?  No.

 

Is the sequence ...?