To find the next term, we need to figure out what is happening from one term to the next.

From 2 to 5, we can see that 3 is added.

From 5 to 14, 9 is added.

From 14 to 41, 27 is added. 

If you look closely, you can notice a trend.

The amount added each time triples. Therefore, the next amount added should be

Thus, 

In the series given,  is added to each previous term to get the next term. Since a fixed number is ADDED each time, this series can be categorized as an arithmetic series. 

First, we need to figure out what the pattern is in this series. Notice how each term results in the following term. In this case, each term is multiplied by  to get the next term. Since each term is MULTIPLIED by a fixed number, this can be defined as a geometric series. 

Common ratio is the number that is multiplied by each term to get the next term in a geometric series. Since the first two terms are  and , we look at what is multiplied between these. Once way to determine this if not immediately obvious is to divide the second term by the first term. In this case we get: 

 which gives us our common ratio. 

This equation is a series in summation notation.

We can see that the bottom k=3 designates where the series starts, 8 represents the stop point, and 1/k represents the rule for summation. We can expand this equation as follows:

Here, we have just substituted "k" for each value from 3 to 8. To solve, we must then find the least common demoninator. That would be 280. This can be found in several ways, such as separating the fractions by like denominators:

We note that there is no common difference between  and  so the sequence cannot be arithmetic.

We also note that there exists a common ratio between two consecutive terms.

Since there exists a common ratio, the sequence is Geometric.

First, let's find a pattern for this sum. Each value has a difference of 3. If we know that the first value is 8, and that k will start at 1, and that each value must go up by 3, we can write the following:

Having determined the the rule for this sum, we can now determine what value it must end at by setting the rule function equal to the last value, 26.

Thus the summation notation can be expressed as follows:

Partial sums (written ) are the first few terms of a sum, so 

If you then just take off the last number in that sum you get the  and so on.

First break down the decimal into a sum of fractions to see the pattern.

and so on.  Thus,

These fractions can be reduced, and the sum becomes

Each term is multiplied by to get the next term which is added. 

For the first 4 terms this would look like

Let  be the index variable in the sum, so if  starts at  the terms in the above sum would look like:

.

The decimal is repeating, so the pattern of addition occurs an infinite number of times.  The sum expressed in sigma notation would then be:

.

In order to evaluate the summation, we must understand what the notation of the expression means:

This sigma notation tells us to sum the values obatined from evaluating the expression at each integer between and including those below and above the sigma. So we're going to start by evaluating the expression at n=1, and then add the value of the expression evaluated at n=2, and so on, until we end by adding the last value of the expression evaluated at n=5. This process is shown mathematically below: