How to find consecutive integers - GRE Quantitative Reasoning
Card 0 of 7
What is the sum of all the integers between 1 and 69, inclusive?
What is the sum of all the integers between 1 and 69, inclusive?
The formula here is sum = average value * number of values. Since this is a consecutive series, the average can be found by averaging only the first and last terms: (1 + 69)/2 = 35.
sum = average * number of values = 35 * 69 = 2415
The formula here is sum = average value * number of values. Since this is a consecutive series, the average can be found by averaging only the first and last terms: (1 + 69)/2 = 35.
sum = average * number of values = 35 * 69 = 2415
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What is the sum of all of the four-digit integers that can be created with the digits 1, 2, 3, and 4?
What is the sum of all of the four-digit integers that can be created with the digits 1, 2, 3, and 4?
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.
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.
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Quantity A: The sum of all integers from 49 to 98 inclusive.
Quantity B: The sum of all integers from 51 to 99 inclusive.
Quantity A: The sum of all integers from 49 to 98 inclusive.
Quantity B: The sum of all integers from 51 to 99 inclusive.
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.
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.
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Quantity A: The sum of all integers from 1 to 30
Quantity B: 465
Quantity A: The sum of all integers from 1 to 30
Quantity B: 465
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 all integers from 1 to 30 can be found using the formula
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If the sum of four consecutive numbers is 38, what is the mean of the largest and the smallest of the four numbers?
If the sum of four consecutive numbers is 38, what is the mean of the largest and the smallest of the four numbers?
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.
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.
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The average of five consecutive integers is 
. What is the largest of these integers?
The average of five consecutive integers is
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:
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
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
Multiply both sides by
Finish solving:
This means that the largest value is:
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The sum of a set of 
consecutive odd integers is 
. What is the fourth number in this set?
The sum of a set of
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 
:
We can represent our numbers as:
Thus, we know:
Solve for
Therefore, the fourth element will be
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