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The sum of k consecutive integers is 41. If the least integer is -40, then k =

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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.
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