Sequences - GRE Quantitative Reasoning

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.

For each quantity, only count the integers that aren't in the other quantity. Both quantities include the numbers 51 to 98, so those numbers won't affect which is greater. Only Quantity A has 49 and 50 (for a total of 99) and only Quantity B has 99. Since the excluded numbers from both quantities equal 99, you can conclude that the 2 quantities are equal.

The sum of all integers from 1 to 30 can be found using the formula

, where is 30. In this case, the sum equals 465.

The sum of 4 consecutive numbers being 38 can be written as the following equation.

We can simplify this to solve for x.

This tells us that the smallest number is 8 and the largest is (8 + 3) = 11. From this, we can find their mean.

There are two ways to figure out this list of integers. On the one hand, you might know that the average of a set of consecutive integers is the "middle value" of that set. So, if the average is and the size , the list must be:

Another way to figure this out is to represent your integers as:

The average of these values will be all of these numbers added together and then divided by . This gives us:

Multiply both sides by :

Finish solving:

This means that the largest value is:

We can represent our numbers as:

will have to be an odd number since the whole sequence is odd. However, this will work out when we do the math. Now, we know that all of these added up will be . We have and the sum of the set , the sum of which is .

Thus, we know:

Solve for :

Therefore, the fourth element will be :

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:

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:

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:

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.

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

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 6th element, continue following this:

The 5th: 47 * 2 + 1 = 95

The 6th: 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

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: