The triangle of numbers created by the famous French Mathematician and philosopher Blaise Pascal which is named after him as Pascalâ€™s Triangle.

**Pascalâ€™s Triangle**

The triangle of numbers created by the
famous French Mathematician and philosopher Blaise Pascal
which is named after him as Pascalâ€™s Triangle. This Pascalâ€™s Triangle
of numbers provides lot of scope to observe various types of number patterns in it.

** **

**Activity**

1. Complete the following
Pascalâ€™s Triangle by observing the number pattern.

**Solution:**

2. Observe the above completed
Pascalâ€™s Triangle and moving the slanting
strips, find the sequence that you see in it and complete them. One is done for
you

(i) 1, 2, 3, 4, 5, 6, 7.

(ii) 1, 3, __6, 10, 15, 21.__

(iii) 1, __4, 10, 20, 35.__

(iv) __1, 5, 15, 35.__

3. Observe the sequence of numbers obtained in
the 3rd and 4th slanting rows of Pascalâ€™s Triangle and find the difference between
the consecutive numbers and complete the table given below.

**Solution:**

** **

__Example 5.2__

Tabulate the 3rd slanting row of the
Pascalâ€™s Triangle by taking the position of the numbers in the slanting row as *x*
and the corresponding values as *y*.

Verify whether the relationship, *y* = [ *x*(*x*+1) ] /2 between *x*
and *y* for the given values is true.

**Solution**

Observe the table carefully. To verify
the relationship between *x* and *y*, let us substitute the values of
*x* and get the values of *y*.

**Think**

The values of* y *are obtained by half of the product of
the two consecutive values of x.

** **

__Example 5.3__

Can
row sum of elements in a Pascalâ€™s Triangle form a pattern?

**Solution**

The row sum of elements of a Pascalâ€™s
Triangle are shown below:

First row = 2^{1}^{âˆ’}^{1} =1

Second row = 2^{2-1}=2 Ã— 1=2

Third row =2^{3-1}=2 Ã— 2=4

Fourth row ==2^{4-1}=2 Ã— 2 Ã—
2=8

Fifth row = 2^{5-1}=2 Ã— 2 Ã— 2 Ã—
2=16

Sixth row = 2^{6-1}=2 Ã— 2 Ã— 2 Ã—
2 Ã— 2=32

Seventh row =
2^{7-1}=2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2=64^{}

Eighth row = 2^{8-1}=2 Ã— 2 Ã— 2 Ã—
2 Ã— 2 Ã— 2 Ã— 2=128

Here *x* denotes the row and *y*
denotes the corresponding row sum. The values of *x* and *y *can be tabulated
as follows:

The relationship between *x* and
*y* is *y* = 2^{x}^{â€“1}_{.}

** **

**DO YOU KNOW**

Observe the pattern obtained
by adding the elements in the slanting rows of the Pascalâ€™s Triangle.

The sequence obtained is
known as Fibonacci
sequence.

** **

**Try these**

**1. Observe the pattern
of numbers given in the slanting rows earlier and complete the Pascalâ€™s Triangle.**

**Solution :**

**2. Complete the given Pascalâ€™s
Triangle. Find the common property of the numbers filled by you. Can you relate
this pattern with the pattern discussed in situation 2. Discuss.**

**Solution :**

** **

__Example 5.4 __

Observe the numbers in the
hexagonal shape given in the Pascalâ€™s
Triangle. The product of the alternate three numbers in
the hexagon is equal to the product of remaining three numbers. Verify this

**Solution**

**Think**

The numbers 1, 3, 6, 10,
... form triangles and are known as triangular numbers. How?

Tags : Information Processing | Term 2 Chapter 5 | 7th Maths , 7th Maths : Term 2 Unit 5 : Information Processing

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7th Maths : Term 2 Unit 5 : Information Processing : Pascalâ€™s Triangle | Information Processing | Term 2 Chapter 5 | 7th Maths

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