To find the sum of the series [tex]\(1 + 2 + 3 + \cdots + 460\)[/tex], you can use the formula for the sum of the first [tex]\(n\)[/tex] natural numbers:
[tex]\[
S = \frac{n(n + 1)}{2}
\][/tex]
where [tex]\(n\)[/tex] is the last number in the series.
In this problem, the series goes up to 460, so [tex]\(n = 460\)[/tex]. Plugging this value into the formula, we get:
[tex]\[
S = \frac{460 (460 + 1)}{2}
\][/tex]
1. First, calculate [tex]\(460 + 1\)[/tex]:
[tex]\[
460 + 1 = 461
\][/tex]
2. Next, multiply 460 by 461:
[tex]\[
460 \times 461 = 211060
\][/tex]
3. Finally, divide the result by 2 to find the sum:
[tex]\[
S = \frac{211060}{2} = 105530
\][/tex]
Thus, the sum of the series [tex]\(1 + 2 + 3 + \cdots + 460\)[/tex] is [tex]\(\boxed{105530}\)[/tex].
