Notice that the series start after the nine.  This means that:

Write the formula to determine the sum of the infinite series.

The first term is:  

The common ratio for each term is:  

Substitute the terms.

The expression becomes:

Simplify the complex fraction.

The terms become:

The answer is:  

A trick to working this is to rewrite the expression 

by, first, factoring the denominator:

Using the method of partial fractions, we can rewrite this as

Comparing numerators, we get 

,

so 

and

and the series can be restated as

This can be rewritten by substituting the integers from 1 to 10, in turn, and adding:

Regrouping, we see this is a telescoping series, in which all numbers after 1 cancel out:

Rewrite the general term so that the sigma notation is as follows:

A series of the form  is an infinite geometric series with common ratio . If  and the initial term is , the sum of the series is 

Here, , , and

Substitute these values in the formula:

Simplifying:

A series of the form  is an infinite geometric series with common ratio . The series converges to a sum if and only if . However, in the given expression, , so , so the series diverges rather than converges.

Rewrite the general term so that the sigma notation is as follows:

A series of the form  is an infinite geometric series with common ratio . If  and the initial term is , the sum of the series is 

Here, , , and

Substitute these values in the formula:

Simplifying:

Rewrite the general term so that the sigma notation is as follows:

A series of the form  is an infinite geometric series with common ratio . If  and the initial term is , the sum of the series is 

Here, , , and

Substitute these values in the formula:

.

A recursive formula creates a sequence where each term is defined by the term(s) that precede it. In other words, in order to know term 12, you have to know term 11, etc.

The problem already tells us that the first term is 2. Let's find the second term.

We continue to find the rest of the terms in this way.

Apply the rule to find the first few terms:

After the sixth term, it is apparent that this cycle will repeat itself, so the first twenty terms of the sequence will be, in order:

Seven of these first twenty terms are positive.

Apply the rule to find the first few terms:

The first positive term of the sequence is .

In order to determine which expression describes the sequence in the problem, we must determine the relationship each entry has with its position in the sequence. For example, for n=1, we must determine which expression involving n will yield a result of 3, for n=2, we must determine which expression will yield a result of 8, and so on, ensuring that the expression holds true for every n value in the sequence. If we check each of our answers, we can see that only the following expression will give the correct result for each increasing value of n:

If we continue, we can see that we will obtain the sequence 3,8,15,24,35,48,63,..., so this expression is the correct representation of the sequence given in the problem.