Question

determine whether each sequence is arithmetic geometric it neither. then find a formula for the nth term of the sequence

Answer 1

The sequence we are given is:

3, 5, 9, 17

and we are asked to find if it is arithmetic, geometric or neither.

To check for arithmetic we observe if there is a common difference between consecutive terms:

a(n+1) - a(n)

5-3 = 2

9-5 = 4

17-9 = 8

So clearly this is NOT an arithmetic sequence

To check for geometric sequence, we study if there is a "common ratio" by looking at the quotient of consecutive terms:

a(n+1) / a(n)

5/3

9/5

17/9

These are all different, therefore this is NOT a geometric sequence either.

Then, we answer NEITHER.

In order to find the term a(n), we notice how we build the different terms:

5 = 3 + 2

9 = 5 + 2^2

17 = 9 + 2^3

so we can try to write the a(n+1) term in terms of the previous one a(n) as:

a(n+1) = a(n) + 2^(n)

this is called a RECURRENT definition of the sequence.