HW Score: $14 \%, 7$ of 50 points
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Legend has it that the great mathematician Carl Friedrich Gauss (1777-1855) at a very young age was told by his teacher to find the sum of the first 100 counting numbers. While his classmates toiled at the problem, Carl simply wrote down a single number and handed the correct answer in to his teacher. The young Carl explained that he observed that there were 50 pairs of numbers that each added up to 101. So the sum of all the numbers must be $50 \cdot 101=5050$. Use the method of Gauss to find the sum.
\[
1+2+3+\ldots+310
\]
\[
1+2+3+\ldots+310=
\]
Answer
\(\boxed{48205}\) is the sum of the first 310 counting numbers.
Step 1 ：Given the problem, we are asked to find the sum of the first 310 counting numbers. We can use the method of Gauss to solve this problem.
Step 2 ：Gauss's method involves pairing numbers in such a way that each pair sums to the same value. In the case of the first 310 counting numbers, we can pair the first and last number, the second and second-to-last number, and so on. Each pair will sum to 311 (1 + 310, 2 + 309, etc.).
Step 3 ：Since there are 310 numbers, there will be \(\frac{310}{2} = 155\) pairs.
Step 4 ：So, the sum of all the numbers will be \(155 \times 311\).
Step 5 ：Calculating this gives us a total sum of 48205.
Step 6 ：\(\boxed{48205}\) is the sum of the first 310 counting numbers.
