I suppose  denotes the n-th term of some sequence, and we're given the 3rd and 5th terms
 denotes the n-th term of some sequence, and we're given the 3rd and 5th terms  and
 and  . On this information alone, it's impossible to determine the 100th term
. On this information alone, it's impossible to determine the 100th term  because there are infinitely many sequences where 2 and 16 are the 3rd and 5th terms.
 because there are infinitely many sequences where 2 and 16 are the 3rd and 5th terms.
To get around that, I'll offer two plausible solutions based on different assumptions. So bear in mind that this is not a complete answer, and indeed may not even be applicable.
• Assumption 1: the sequence is arithmetic (a.k.a. linear)
In this case, consecutive terms <u>d</u>iffer by a constant d, or

By this relation,

and by substitution,

We can continue in this fashion to get


and so on, down to writing the n-th term in terms of the first as

Now, with the given known values, we have


Eliminate  to solve for d :
 to solve for d :

Find the first term  :
 :

Then the 100th term in the sequence is

• Assumption 2: the sequence is geometric
In this case, the <u>r</u>atio of consecutive terms is a constant r such that

We can solve for  in terms of
 in terms of  like we did in the arithmetic case.
 like we did in the arithmetic case.

and so on down to

Now,


Eliminate  and solve for r by dividing
 and solve for r by dividing

Solve for  :
 :

Then the 100th term is

The arithmetic case seems more likely since the final answer is a simple integer, but that's just my opinion...