The least integer of a set of consecutive integers is -25 if the sum of these integers is 26,how many integers are in this set?
Let's start by setting up the problem. We know that the least integer in the set is -25 and the sum of the integers is 26. We can represent this set of consecutive integers with an arithmetic sequence.
Let's assume the number of integers in the set is 'n'.
The sum of an arithmetic sequence can be calculated using the formula:
Sum = (n/2) * (first term + last term)
In this case, the first term is -25, and the last term can be found by adding (n - 1) to the first term.
Substituting these values into the formula, we have:
26 = (n/2) * (-25 + (-25 + (n - 1)))
Now, let's solve for 'n':
26 = (n/2) * (-25 + (-25 + n - 1))
26 = (n/2) * (-25 + n - 26)
26 = (n/2) * (n - 51)
52 = n^2 - 51n
n^2 - 51n - 52 = 0
Now, we can solve this quadratic equation for 'n' by factoring or using the quadratic formula.
Factoring:
(n - 52)(n + 1) = 0
Setting each factor equal to zero, we can find the possible values of 'n':
n - 52 = 0 or n + 1 = 0
n = 52 or n = -1
Since the number of integers cannot be negative, we discard the solution 'n = -1'.
Therefore, there are 52 integers in the set.
To find the number of integers in the set, we can use a formula for the sum of consecutive integers.
First, let's define the variables:
n = the number of integers in the set
x = the least integer in the set
We know that the sum of the consecutive integers is 26. Using the formula for the sum of consecutive integers, we can write the equation:
Sum = (n/2) * (2x + (n-1))
Substituting the given values into the equation, we have:
26 = (n/2) * (2*(-25) + (n-1))
Now we can solve this equation:
26 = (n/2) * (-50 + n - 1)
26 = (-50n + n^2 - n)/2
52 = -50n + n^2 - n
0 = n^2 - 51n + 52
Since we are looking for a positive number of integers, we can solve this quadratic equation. Factoring it will give us:
0 = (n - 1)(n - 52)
So, n - 1 = 0 or n - 52 = 0
If n - 1 = 0, then n = 1
If n - 52 = 0, then n = 52
Since we are dealing with a consecutive set of integers, it is not possible to have just one integer with a sum of 26. Therefore, the number of integers in the set is 52.