Prime numbers are numbers that can only divided by one and themselves.  Breaking 60 into its prime factors yields:

Do NOT try to add all of these.  It is key that you notice the pattern.  Begin by looking at the first and the last elements: 1 and 99.  They add up to 100.  Now, consider 3 and 97.  Just as 1 + 99 = 100, 3 + 97 = 100.  This holds true for the entire list.  Therefore, it is crucial that we find the number of such pairings.

1, 3, 5, 7, and 9 are paired with 99, 97, 95, 93, and 91, respectively.  Therefore, for each 10s digit, there are 5 pairings, or a total of 500.  To get all the way through our numbers, you will have to repeat this process for the 10s, 20s, 30s, and 40s (all the way to 49 + 51 = 100).

Therefore, there are 500 (per pairing) * 5 pairings = 2500.

With series, you can always "walk through" the values to find your answer. Based on our equation, we can rewrite  as :

You then continue for the third and the fourth element:

When you are asked to find elements in a series that are far into its iteration, you need to find the pattern. You absolutely cannot waste your time trying to calculate all of the values between  and . Notice that for every element after the first one, you subtract . Thus, for the second element you have:

For the third, you have:

Therefore, for the 40th and 70th elements, you will have:

The sum of these two elements is:

The first term and second term average out to 6. So the third term is 6. Now add 6 to the preceding two terms and divide by 3 to get the average of the first three terms, which is the value of the 4th term. This, too, is 6 (18/3)—all terms after the 2nd are 6, including the 39th. Thus, the answer is 6.

First, consider the change in each element.  Notice that in each case, a given element is twice the preceding one plus one:

11 = 2 * 5 + 1

23 = 11 * 2 + 1

47 = 23 * 2 + 1

To find the 6^th element, continue following this:

The 5^th: 47 * 2 + 1 = 95

The 6^th: 95 * 2 + 1 = 191

The first term of the sequence is , so here , and we're interested in finding the 20th term, so we'll use n = 20.

Plugging these values into the given expression for the nth term gives us our answer.

and

Our sequence begins as 1, 2.

Element 3: (Element 1 + Element 2) * 3 = (1 + 2) * 3 = 3 * 3 = 9

Element 4: (Element 2 + Element 3) * 3 = (2 + 9) * 3 = 11 * 3 = 33

Element 5: (Element 3 + Element 4) * 3 = (9 + 33) * 3 = 42 * 3 = 126

Let us first write the value of a consecutive term in a numerical format:

Consequently,

Using the first equation, we can define  in terms of :

This allows us to rewrite

as

Rearrangement of terms allows us to solve for :

Now, using our second equation, we can find , the first term:

Begin by interpreting the general definition:

This means that every number in the sequence is five greater than the element preceding it.  For instance:

It is easiest to count upwards: