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When we have a given number k, the consecutive number to k will be (k + 1), the next consecutive will be (k + 2), and so on.
With this, we can find that the value of n is 100.
So here we have n consecutive numbers, let's assume we start with the number k, then the sum will be:
k + (k + 1) + ... + (k + n - 1)
Now we know that this is 100 less than the sum of the next n consecutive numbers.
For the next n consecutive numbers we need to start at k + 1, then we get:
(k + 1) + (k + 2) + ... + (k + n - 1) + (k + n)
If we take the difference we will get:
((k + 1) + (k + 2) + ... + (k + n - 1) + (k + n)) - (k + (k + 1) + ... + (k + n - 1)) = 100
As you can see, there are a lot of terms that are in both parentheses, then when we do that difference a lot of them cancel out.
So we will et:
((k + 1) + (k + 2) + ... + (k + n - 1) + (k + n)) - (k + (k + 1) + ... + (k + n - 1))
= (k + n) - k = k + n - k = n
And we knew that this difference must be equal to 100, then we have:
n = 100
We just found the value of n.
If you want to learn more, you can read:
https://brainly.com/question/1767889