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Pascal's triangle

Do you remember the first binomial formula:

$$ (a + b) ^ 2 = a ^ 2 + 2ab + b ^ 2 $$?

Let's think a little further:
$$ (a + b) ^ 0 $$
$$ (a + b) ^ 1 $$
$$ (a + b) ^ 2 $$
$$ (a + b) ^ 3 $$
$$…$$

What is the result for this series?
$$ (a + b) ^ 0 = 1 $$
$$ (a + b) ^ 1 = 1 * a + 1 * b $$
$$ (a + b) ^ 2 = 1 * a ^ 2 + 2 * ab + 1 * b ^ 2 $$
$$ (a + b) ^ 3 = 1 * a ^ 3 + 3 * a ^ 2b + 3 * ab ^ 2 + 1 * b ^ 3 $$
$$ (a + b) ^ 4 =… $$

The one before the individual summands
let's take a closer look now

$$ * a + $$$$ * b $$
$$ * a ^ 2 + $$$$ * ab + $$$$ * b ^ 2 $$
$$$$$$ * a ^ 3 + $$$$ * a ^ 2b + $$$$ * from ^ 2 + $$$$ * b ^ 3 $$
$$ …$$
and write them down in the form of a triangle:
       
         
          
            
$$…$$

About the creation of the triangle

       
          
             

You can tell from the first three lines
detect:

  • There is always a 1 on the outside of each line.
  • A coefficient in a row follows by adding the two coefficients in the row above.
Blaise Pascal (1623--1662)

The triangle named after Pascal was known more than 1000 years ago. But he was the first to systematically investigate it.

If these two rules are applied, you will get, for example, the following lines from the first three lines:

The Pascal triangle

Now you can continue to apply the rules and you get the following scheme of Pascal's triangle:

If you feel like you can add more lines.

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Special features of the Pascal triangle (1)

The line total

If you add up all the numbers in a row, you get a sequence of numbers - note that the 1st row as Line 0 referred to as:
Line 0: $$ 1 = 1 $$
Line 1: $$ 1 + 1 = 2 $$
Line 2: $$ 1 + 2 + 1 = 4 $$
Line 3: $$ 1 + 3 + 3 + 1 = 8 $$
Line 4: $$ 1 + 4 + 6 + 4 + 1 = 16 $$
$$…$$

You definitely realize that the sum of the numbers increases from line to line doubled. If you are in Pascal's triangle as Index $$ n $$ If you choose the exponent of the binomial $$ (a + b) $$, you can see the general law of formation for the sum $$ S $$ of the numbers from the following scheme:

If $$ n $$ is the exponent of the binomial $$ (a + b) $$, then the image law for the line sum is $$ S $$ of the numbers $$ S = 2 ^ n $$.
Examples:
$$ 2 ^ 0 = 1 $$ (note the definition: every number raised to the power of $$ 0 $$ results in $$ 1 $$)
or
$$2^3 = 2 * 2 * 2 = 8$$

Special features of the Pascal triangle (2)

Many ways lead to the goal

Consider the $$ 1 $$ in the first box of the triangle from above as Starting point. Now count the distances from "top to bottom" to the field with the $$ 2 $$. You can only do two shortest Because of getting there.

The figure above shows you that there are exactly $$ 4 $$ shortest paths from the starting point $$ 1 $$ to the field with the $$ 4 $$.
Try other goals! You will find that this always applies.

Special features of the Pascal triangle (3)

Divisibility pattern of numbers

The numbers in Pascal's triangle are now marked, which just are - i.e. all numbers divisible by $$ 2 $$. This representation is a little different - and maybe looks prettier!

Apparently all triangles are created, which are oriented the other way round to the original triangle.

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Special features of the Pascal triangle (4)

Square numbers

Look at the shape of the numbers of Pascal's triangle opposite. Note the third column with the sequence of numbers
$$1, 3, 6, 10, 15, …$$
If you add up any two consecutive numbers, so
$$ 1 + 3 $$ or $$ 3 + 6 $$ or $$ 10 + 15 $$, you get a square number.

Another way of representing the numbers of Pascal's triangle is the following:
$$1$$
$$1 1$$
$$1 2 1$$
$$1 3 3 1$$
$$1 4 6 4 1$$
$$1 5 10 10 5 1$$
$$1 6 15 20 15 6 1$$

Fibonacci numbers


Look at the numbers marked by the diagonals and form the sum of each. A sequence of numbers is created again, the so-called Fibonacci sequence:
$$1, 1, 2, 3, 5, 8, …$$.
Each Fibonacci number is the sum of the two preceding Fibonacci numbers. This sequence of numbers also has a multitude of relationships with other areas of mathematics. Find out more about this series of numbers on the Internet.

There are many other peculiarities of Pascal's triangle. Maybe there is something else in the exercises - let yourself be surprised!