There is a simple and innovative presentation of the difference of the harmonic series (which we talked about last week in relation to the group of cantilevered books), that is, the total 1 + 1/2 + 1/3 + 1/4 + 1/5 … grows indefinitely with increasing Number of additions. Growth is very slow (requires parts of additions to reach 100), but not specific.

Effective way:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 … it is clearly greater than

1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 … = 1 + 1/2 + 1/2 + 1/2 …

And since the second series grows indefinitely, so does the first series, all of whose corresponding terms are greater than or equal to, as well. The rendition is due to the great medieval mathematician, physicist, and astronomer Nicolas de Orsme, who predicted Copernicus two centuries earlier by saying that it was the Earth that moves, not the sun and the stars. He did so with caution enough to save himself from the stake, which is why the credit goes to Copernicus and Galileo.

For the three hundredth number, or what is the same, the sum of the first 100 natural numbers, little Gauss found it in a few seconds when he realized that 1 + 100 = 2 + 99 = 3 + 98 = 4 + 97 … = 101, so this sum is 50 pairs of numbers that add up to 101, i.e. 50 x 101 = 5050.

Triangular numbers are so named because they can be represented as groups of points arranged in such a way that they form an equilateral triangle, with the points on each side indicating the arrangement in the triangle’s sequence (with 1 being included as the first term). Called the Pythagoreans *Rubaiyat* To represent 10 as equilateral triangle points; A very familiar configuration to us, since it is equivalent to the usual arrangement of bolts for bowling in *Bowling*.

Note that 1 + 3 = 4 = 2², 3 + 6 = 9 = 3², 6 + 10 = 16 = 4²… Will the sum of two consecutive triangles always be a perfect square?

**Full square**

But why do we call the power 2 of a number that is the square of that number? Well, for the same reason we call the numbers we just saw triangles: because perfect squares or square numbers can be represented as sets of points arranged in such a way that they form a square. Since 1 is a perfect square, since 1 square = 1, in this case it is included as the first term in the series *Of course*As the French say.

Thus, the square numbers are 1, 4, 9, 16, 25, 36 …

Obviously, the square number n is n², and n² is equal to the sum of the first n odd numbers:

2² = 1 + 3

3² = 1 + 3 + 5

4² = 1 + 3 + 5 + 7

5² = 1 + 3 + 5 + 7 + 9

Why is this the case for any square number?

A perfect square can end in 0, 1, 4, 5, 6, and 9, but not 2, 3, 7, or 8. This can be easily verified by squaring all 10 numbers, since the square of the number ends with the same square of the number the last one.

Another property of perfect squares is that they always contain an odd number of divisors. why?

**Carlo Frappetti ***Writer, mathematician and member of the New York Academy of Sciences. He has published more than 50 popular scholarly works for adults, children and teens, including “Damn Physics,” “Damn Mathematics,” and “The great game”. He was the screenwriter of “La bola de cristal”.*

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