The sum of first n positive integers

The sum of first n positive integers

Σi represents the sum of first n positive integers, 1 + 2 + 3 + . . . + n,

Show that Σ i = n(n+1)/2

Step 1. By definition, Σi² = 1² + 2² + 3² + 4² + . . . + n² --> eq. (1)

Step 2. Also, Σ (i + 1)² = 2² + 3² + 4² + . . . + n² + (n+1)² --> eq. (2)

Step 3. If we take the difference: (2) – (1), Σ (i + 1)² - Σ i² we get,

= [2²+3²+4²+. . .+n²+(n+1)²] - [1²+2²+3²+4²+. . .+n²]

Step 4. Simplifying, we get Σ (i + 1)² - Σ i² = (n+1)² – 1²

Step 5. Expanding (i + 1)² and (n+1)² we get,

Σ(i²+2i+1) - Σi² = n²+2n+1–1

Step 6. Using Summation properties, we get
Σ i² + Σ 2i + Σ 1 - Σ i² = n² + 2n

Step 7. Simplifying both sides of the equation, we get,
2Σ i + n = n² + 2n

Step 8. Solving for Σ i , we get Σ i = n(n+1)/2

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