To find the value of k, the number of consecutive integers, given that the sum is 41 and the least integer is -40, you can set up an equation.
Let's call the consecutive integers starting from the least integer -40 as -40, -39, -38, ..., and so on.
The sum of k consecutive integers can be represented as:
Sum = (-40) + (-39) + (-38) + ... + (-38 + k)
Since the sum is given as 41, you can write the equation:
41 = (-40) + (-39) + (-38) + ... + (-38 + k)
Now, you can simplify this equation:
41 = (-40 - 39 - 38 - ... - (38 - k))
Combine the negative terms:
41 = -(40 + 39 + 38 + ... + (38 - k))
Now, you need to find the sum of consecutive integers from -40 up to -(40 - k + 1).
The sum of consecutive integers can be found using the formula for the sum of an arithmetic series:
Sum = (n/2) * [2a + (n - 1)d]
Where:
- n is the number of terms,
- a is the first term,
- d is the common difference.
In this case, a = -40, and you are finding the sum of integers from -40 up to -(40 - k + 1), so the last term is -(40 - k + 1), and the common difference is 1.
Now, you can plug these values into the formula:
41 = (n/2) * [2 * (-40) + (n - 1) * 1]
Simplify further:
41 = (n/2) * (-80 + n - 1)
41 = (n/2) * (-79 + n)
Now, you want to solve for n (the number of terms). You can start by multiplying both sides by 2 to get rid of the fraction:
82 = n * (-79 + n)
Now, expand and rearrange:
82 = -79n + n^2
Combine like terms:
0 = n^2 - 79n + 82
Now, you have a quadratic equation. You can factor it or use the quadratic formula to solve for n. Factoring doesn't yield integer values for n in this case, so let's use the quadratic formula:
n = (-b ± √(b² - 4ac)) / (2a)
In this equation, a = 1, b = -79, and c = 82. Plug these values into the formula:
n = (-(-79) ± √((-79)² - 4 * 1 * 82)) / (2 * 1)
n = (79 ± √(6241 - 328)) / 2
n = (79 ± √(5913)) / 2
Since you're interested in a positive integer value for k, you'll take the positive square root:
n = (79 + √(5913)) / 2 ≈ 45.49
Since n represents the number of terms, and k represents the difference between the highest and lowest integer, k = n - 1:
k = 45.49 - 1 ≈ 44.49
However, k must be a whole number since it represents the number of consecutive integers. Therefore, k is approximately 44 (rounded down to the nearest whole number).
So, k ≈ 44.