To find the sum [tex]\(1 + 2 + 3 + \ldots + 270\)[/tex] using the method of Carl Friedrich Gauss, we can recognize the pattern and symmetry within the sequence.
### Steps:
1. Identify the sequence and number of terms:
The sequence given is [tex]\(1, 2, 3, \ldots, 270\)[/tex]. This sequence is an arithmetic series with:
- The first term ([tex]\(a_1\)[/tex]) = 1
- The last term ([tex]\(a_n\)[/tex]) = 270
- The total number of terms ([tex]\(n\)[/tex]) = 270
2. Pair the numbers:
Gauss observed that by pairing the first term with the last term, the second term with the second-last term, and so on, each pair would sum to a constant value.
For our sequence, we have pairs such as:
[tex]\[
(1 + 270), (2 + 269), (3 + 268), \ldots, (135 + 136)
\][/tex]
3. Calculate the sum of each pair:
Notice that each such pair sums to:
[tex]\[
1 + 270 = 271, \quad 2 + 269 = 271, \quad 3 + 268 = 271, \quad \ldots, \quad 135 + 136 = 271
\][/tex]
So, each pair has a total sum of [tex]\(271\)[/tex].
4. Determine the number of pairs:
As [tex]\(270\)[/tex] terms are paired, we can divide the total number of terms by 2 to get the number of pairs:
[tex]\[
\frac{270}{2} = 135
\][/tex]
5. Compute the total sum:
The total sum of the sequence is the number of pairs multiplied by the sum of each pair:
[tex]\[
135 \times 271 = 36585
\][/tex]
Thus, the sum of the sequence from 1 to 270 is:
[tex]\[
\boxed{36585}
\][/tex]
