To solve the problem of finding the sum (1 + 2 + 3 + \dots + 100), we use the properties of arithmetic sequences and a clever pairing method:

Step 1: Define the sum

Let (S = 1 + 2 + 3 + \dots + 99 + 100).

Step 2: Reverse the sum

Write the sum backward:
(S = 100 + 99 + 98 + \dots + 2 + 1).

Step 3: Add the two versions of (S)

Add the forward and reversed sums:
(2S = (1+100) + (2+99) + (3+98) + \dots + (99+2) + (100+1)).

Step 4: Calculate the total of paired terms

Each pair ((k + (101 - k)) = 101), and there are 100 such pairs (since there are 100 terms):
(2S = 100 \times 101).

Step 5: Solve for (S)

Divide both sides by 2:
(S = \frac{100 \times 101}{2} = 50 \times 101 = 5050).

Answer: (\boxed{5050})

作者声明:本文包含人工智能生成内容。