5 visual demonstrations

For secondary.

Passing through the blog of Bill the Lizard saw some very interesting shows I want to share:

Demo # 1:

The sum of n odd numbers is n ^2.

1 + 3 + 5 + ... + (2n - 1) = n ²

This is very clear 1 + 3 + 5 + ... arranged in the order shown form a square. The area is covered by n ^2.

Demonstration No. 2:

The sum of n consecutive integers is n (n +1) / 2.

1 + 2 + 3 + ... + n = n. (N +1) / 2

If you see the image we have a rectangle of width n height n +1. If the sum of 1 + 2 + 3 ... is half the area of the rectangle then the answer of this is n (n +1) / 2.

Demo # 3:

The sum of fractions whose denominators powers of 2 results 1.

1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ... = 1

It is easy to see that the result we leave a

Demo N ° 4:

The sum of fractions whose denominator powers of 3 returns 1 / 2.

1 / 3 + 1 / 9 + 1 / 27 + 1 / 81 + ... = half

If we translate those parts of the green shaded areas of gray side, half are shaded square.

Demo N ° 5:

The sum of fractions whose denominator powers of 4 returns 1 / 3.

1 / 4 + 1 / 16 + 1 / 64 + 1 / 256 + ... = 1 / 3

References:

BILL THE LIZARD [online]. Six Visual Proofs [Date accessed: August 9, 2009]. Available at: http://www.billthelizard.com/2009/07/six-visual-proofs_25.html

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