मुफ्त योगफल कैलकुलेटर
श्रृंखला का योगफल (Σ) निकालें। सूत्र और सीमाएं दर्ज करें और कुल योग की गणना करें।
Requested
385
Sum Integers55
Sum Squares385
Sum Cubes3,025
सूत्र
Summation Formulas
Sum of Integers
sum(i, 1, n) = n(n+1)/2
Sum of Squares
sum(i², 1, n) = n(n+1)(2n+1)/6
Sum of Cubes
sum(i³, 1, n) = [n(n+1)/2]²
Notice that the sum of cubes equals the square of the sum of integers.
हल किया गया उदाहरण
Sum of i² from 1 to 10.
- 01Sum of integers: 10×11/2 = 55
- 02Sum of squares: 10×11×21/6 = 2310/6 = 385
- 03Sum of cubes: (55)² = 3025
अक्सर पूछे जाने वाले प्रश्न
What is a summation?
A summation (denoted by the sigma symbol) adds up a sequence of terms following a pattern. For example, sum from i=1 to 5 of i² = 1+4+9+16+25 = 55.
Why does the sum of cubes equal the square of the sum?
This is a beautiful identity: 1³+2³+...+n³ = (1+2+...+n)². It can be proven algebraically by induction or geometrically by arranging squares.
Are there formulas for higher powers?
Yes, Faulhaber's formulas give closed-form expressions for the sum of any fixed power k. However, they become increasingly complex for larger k.
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