is a simple sum of natural numbers from 1 to 10: you can get the result
If you use simple geometric principles to express, each Numbers can be represented by unit squares, 1 square represents 1, 2 squares represent 2, and 3 squares represent 3........, then the area of the square is
But the calculation above. Half of the n squares are missing, so 10/2=5 is added. The result is the sum of natural numbers from 1 to 10
Then for all natural numbers from 1 to 100 What is the sum equal to? Similarly, using the above method, the result is 5050, which is exactly what the genius mathematician Gauss got when he was 8 years old
We raise it to a height and calculate the sum of the squares of all natural numbers from 1 to n
We also use geometric methods to describe this principle, but replace the unit square with In the unit cube, the first layer represents 1^2, the second layer represents 2^2, and the third layer represents 3^2........
The above figure is similar to a quadrangular pyramid. The volume formula of the pyramid is n^/3
Don’t worry, we imitate geometric principles, the first represents a cube with side length 1, the second represents a cube with side length 2, and the third represents A cube with a side length of 3...... The last cube with a side length of n forms a super cube.
With reference to the above geometric principles, the volume of this super cube is n^4/4,
The derivation of the above series is fine. It is deduced that the sum of the k powers from 1 to n is approximately equal to
which corresponds to the integral formula of x^k
So the mystery of mathematics is worth exploring, because you will get unexpected results
