Sure! Let's consider the given series:
[tex]\[ S = 1 - 2 + 3 - 4 + 5 - 6 + \cdots + 2019 - 2020 + 2021. \][/tex]
We'll break this down step by step.
1. Group the terms in pairs: Each pair will consist of two consecutive terms, an odd and the next even number.
The series can be rewritten as:
[tex]\[
S = (1 - 2) + (3 - 4) + (5 - 6) + \cdots + (2019 - 2020) + 2021
\][/tex]
2. Simplify each grouped pair:
Observe that:
[tex]\[
1 - 2 = -1, \quad 3 - 4 = -1, \quad 5 - 6 = -1, \quad \cdots, \quad 2019 - 2020 = -1
\][/tex]
Each group of two terms contributes [tex]\(-1\)[/tex] to the total sum.
3. Count the number of pairs:
Since the numbers range from 1 to 2020 (an even number of terms), there are [tex]\(2020 / 2 = 1010\)[/tex] pairs.
4. Sum up the contributions of the pairs:
Each of the 1010 pairs contributes [tex]\(-1\)[/tex], so the total contribution from the pairs is:
[tex]\[
1010 \times (-1) = -1010
\][/tex]
5. Add the remaining term:
After 2020, we have one remaining term, which is [tex]\(+2021\)[/tex].
6. Combine all contributions:
The complete sum is:
[tex]\[
S = -1010 + 2021 = 1011
\][/tex]
Thus, the value of the series [tex]\(1 - 2 + 3 - 4 + \cdots + 2019 - 2020 + 2021\)[/tex] is:
[tex]\[
\boxed{1011}
\][/tex]
