1+(-2)+(-3)+4+5+(-6)+(-7)+8+9+...+(-2021)+(-2022)

To find the sum of the given sequence, we can group the positive and negative terms separately.

Grouping the positive terms:
4 + 5 + 8 + 9 + ... + 2020 + 2021 + 2022

This is an arithmetic sequence with a common difference of 1. We can use the formula for the sum of an arithmetic sequence:

Sn = (n/2)(2a + (n-1)d)

where Sn is the sum of the sequence, n is the number of terms, a is the first term, and d is the common difference.

In this case, a = 4, d = 1, and the last term is 2022. We need to find the number of terms, n.

2022 = 4 + (n-1)(1)
2022 - 4 = n - 1
2018 = n - 1
2019 = n

Now we can substitute these values into the formula:

Sn = (n/2)(2a + (n-1)d)
Sn = (2019/2)(2(4) + (2019-1)(1))
Sn = (2019/2)(8 + 2018)
Sn = (2019/2)(2026)
Sn = 2043237

So the sum of the positive terms is 2,043,237.

Grouping the negative terms:
-2 + (-3) + (-6) + (-7) + ... + (-2020) + (-2021) + (-2022)

This is also an arithmetic sequence with a common difference of -1. We can use the same formula to find the sum.

In this case, a = -2, d = -1, and the last term is -2022. We need to find the number of terms, n.

-2022 = -2 + (n-1)(-1)
-2022 + 2 = n - 1
-2020 = n - 1
-2019 = n

Now we can substitute these values into the formula:

Sn = (n/2)(2a + (n-1)d)
Sn = (-2019/2)(2(-2) + (-2019-1)(-1))
Sn = (-2019/2)(-4 + 2020)
Sn = (-2019/2)(2016)
Sn = -2031612

So the sum of the negative terms is -2,031,612.

Finally, we can find the sum of the entire sequence by adding the sums of the positive and negative terms:

2,043,237 + (-2,031,612) = 11,625

Therefore, the sum of the given sequence is 11,625.