Page 347 - Math_Genius_V1.0_C8_Flipbook

E:\Working\Focus_Learning\Math_Genius-8\Open_Files\20_Chapter_15\Chapter_15
\ 06-Jan-2025 Surendra Prajapati Proof-6 Reader’s Sign _______________________ Date __________

Example 9: Find the missing terms in the following number
sequence: 10, 13, …, …, 22. Quick Check

Solution: The missing terms are 16 and 19, as the succeeding term What number comes next to
each number sequence?
is 3 more than the preceding term. 1. 81, 27, 9, 3, …
So, the number sequence is 10, 13, 16, 19, 22. 2. 7, 12, 18, 25, 33, 42, …
3. 0, 11, 22, …, 44, …, 66.
Pascal’s Triangle

Pascal’s triangle is a special triangle that is named after Blaise Pascal. Pascal’s triangle is an
arrangement of numbers in a triangular array such that the numbers at the end of each row are 1
and the remaining numbers are the sum of the nearest two numbers in the above row.

1 ‘1’s 1
1 1 Counting numbers 1 1

1 2 1 Triangular numbers 1 2 1
1 3 3 1 1 3 3 1

1 4 6 4 1 1 4 6 4 1

We observe the following number patterns in the Pascal’s
triangle: Think and Answer
• The left most and the right most diagonal contain the Find the missing number in the
number 1. given pattern.
1
• The second diagonal from the left contains counting 1 1
numbers. 1 2 1
• The third diagonal from the left contains triangular 1 3 1
numbers. 1 4 1
1 5 10 1
• Each row of triangle represents the corresponding 1 1
exponent of 11 starting from 0.
3
1
0
For example, row 1 = 11 = 1, row 2 = 11 = 11, row 3 = 11 = 121, row 4 = 11 = 1331, and so on.
2
Magic Squares
A magic square is an arrangement of distinct numbers (i.e., each number 6 7 2 15
is used once), usually integers arranged in an n × n table such that the sum
of numbers in each row, in each column, and in each diagonal is same. This 1 5 9 15
sum is called magic sum or magic constant. 8 3 4 15

Look at the adjacent table. This is a 3 × 3 magic square. In this magic square, 15 15 15 15 15
the magic sum is 15.
Here, the number 5 is the ‘central number’ of the given magic square whose magic sum/constant
is 15.

There is relationship between the magic sum and the central number of n × n magic square. That is,
Magic sum/constant = Order × Central number.

345 Playing With Numbers