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:

For this problem, you definitely do not want to "count upwards" to the full value of the sequence.  Therefore, the best approach is to consider the general pattern that arises from the general definition:

This means that for every element in the list, each one is  greater than the one preceding it.  For instance:

Now, notice that the first element is:

The second is:

The third could be represented as:

And so forth...

Now, notice that for the third element, there are only two instances of .  We could rewrite our sequence:

This value will always "lag behind" by one.  Therefore, for the st element, you will have:

For this problem, you definitely do not want to "count upwards" to the full value of the sequence.  Therefore, the best approach is to consider the general pattern that arises from the general definition:

This means that for every element in the list, each one is  less than the one preceding it.  For instance:

Now, notice that the first element is:

The second is:

The third could be represented as:

And so forth...

Now, notice that for the third element, there are only two instances of .  We could rewrite our sequence:

This value will always "lag behind" by one.  Therefore, for the th element, you will have:

For this sequence, you do not have a starting point (i.e. ); however, you are able to interpret it relatively easily. The sequence is merely one in which each number is twenty larger than the one preceding it. Therefore, if  were , you would have:

Now, to find difference between the 20th and the 30th element, it is merely necessary to count the number of twenties that would be added for each of those elements. For instance, the difference between the 21st and the 20th elements is . Thus, since you would add a total of ten twenties from the 20th to the 30th element, you know that the difference between these two values is .

Since this sequence is linear, we know that it will add the same amount for each element. This means that you can evenly divide the difference between the first and the ninth term. Be careful! There will be eight total increases between these terms. (Think this through: 1 to 2, 2 to 3, 3 to 4, etc.)

Thus, we know that the total difference between these terms is:

Now, dividing this among the eight increases that happen, we know:

This means that for each element, we add  to the one prior to it. This means that our general sequence is defined as:

First we need to find out how many possible numbers there are. The number of possible four-digit numbers with four different digits is simply 4 * 4 * 4 * 4 = 256.  

To find the sum, the formula we must remember is sum = average * number of values. The last piece that's missing in this formula is the average. To find this, we can average the first and last number, since the numbers are consecutive. The smallest number that can be created from 1, 2, 3, and 4 is 1111, and the largest number possible is 4444. Then the average is (1111 + 4444)/2.

So sum = 5555/2 * 256 = 711,040.