What is the sum of all integers x,such -36 <x<38?
pair them up in this way:
-35,35
-34,34
...
The sum of all those pairs is zero. So, which numbers are left over?
Add them up.
Ah, the sum of all integers between -36 and 38? Well, I could count them all, but that sounds rather tedious. So, instead, I'll just use my trusty calculator to add them up. *beep boop beep* Ah, it seems like the sum is zero! You see, for every positive number, there's a corresponding negative number that cancels it out. It's like the universe is playing a big game of addition and subtraction with itself! Isn't math marvelous?
To find the sum of all integers x such that -36 < x < 38, we can use the formula for the sum of an arithmetic series.
An arithmetic series is characterized by a first term (a), a last term (l), and a common difference (d), where the terms form a sequence that increases or decreases by the same amount in each step.
In this case, our first term (a) is -35 (since -36 is not included), our last term (l) is 37 (since 38 is not included), and our common difference (d) is 1.
We can now use the formula for the sum of an arithmetic series, which is given by:
S = (n/2) * (a + l)
Where S represents the sum of the series, n represents the number of terms, a represents the first term, and l represents the last term.
To find the number of terms (n), we can subtract the first term from the last term and add 1 (since both terms are included in the series):
n = l - a + 1
Plugging in the values, we have:
n = 37 - (-35) + 1
n = 37 + 35 + 1
n = 73
Now we can substitute the values of n, a, and l into the formula for the sum:
S = (73/2) * (-35 + 37)
S = (73/2) * 2
S = 73 * 1
S = 73
Therefore, the sum of all integers x such that -36 < x < 38 is 73.
To find the sum of all integers within a given range, you need to sum up all the integers from the starting point of the range to the ending point.
In this case, the range is defined as -36 < x < 38.
First, let's find the sum of all integers from -36 to 37:
-36 + (-35) + (-34) + ... + 0 + 1 + 2 + 3 + ... + 37
In this sequence, you can notice that there are equal numbers of positive and negative integers. Therefore, you can pair each positive integer with its corresponding negative integer. For example, -36 + 36 = 0, -35 + 35 = 0, and so on.
The only integer left alone is 0, which does not have a pair.
So the sum can be simplified as follows:
(0) + (0) + (0) + ... + (0) = 0
Therefore, the sum of all integers x such that -36 < x < 38 is 0.