Concept 9 of 19Foundation
Watch on YouTubeVideoKey sum formulas you'll use again and again
Memorize these five sum formulas:
- 1 + 2 + 3 + … + n = n(n+1)/2
- 1 + 3 + 5 + … + (2n−1) = n² (sum of first n odd numbers)
- 2 + 4 + 6 + … + 2n = n(n+1) (sum of first n even numbers)
- 1² + 2² + 3² + … + n² = n(n+1)(2n+1)/6
- 1³ + 2³ + 3³ + … + n³ = [n(n+1)/2]²
Stunning fact: sum of cubes = (sum of naturals)². So 1+8+27+64 = 100 = 10² = (1+2+3+4)². Magic.
Example
1+2+…+100 = 100·101/2 = 5050.
2+4+…+50 = 25·26 = 650 (here n=25).
Sum of first 10 cubes = 55² = 3025.
2+4+…+50 = 25·26 = 650 (here n=25).
Sum of first 10 cubes = 55² = 3025.
💡 Tip:Sum of evens = n(n+1), sum of odds = n². They're siblings — don't swap them.
▸Why does this work? (derivation)
Why 1+2+…+n = n(n+1)/2 (the Gauss trick)?
Write: 1 + 2 + 3 + … + n
And: n + (n−1) + (n−2) + … + 1
Pair them: each column sums to (n+1); there are n columns → 2·Sum = n(n+1) → Sum = n(n+1)/2. ✓
Why odd sum = n²? Stack n rows of dots, each row one longer odd number. They form a perfect n×n square → n² dots.
Write: 1 + 2 + 3 + … + n
And: n + (n−1) + (n−2) + … + 1
Pair them: each column sums to (n+1); there are n columns → 2·Sum = n(n+1) → Sum = n(n+1)/2. ✓
Why odd sum = n²? Stack n rows of dots, each row one longer odd number. They form a perfect n×n square → n² dots.
🎯 Try it!
5 questions to check what you just read.
0 / 5
- Q1.Sum of first 20 natural numbers?
- Q2.Sum of first 10 odd numbers?
- Q3.Sum 2 + 4 + 6 + … + 40?
- Q4.Sum of first 4 cubes (1³+2³+3³+4³)?
- Q5.Find n if 1 + 2 + … + n = 36.