complete the given Patten
1 square+2square+2square
Answers
Answer:
The difference between consecutive squares is always an odd number. If n is any natural number then the difference between the square of n and the square of the next natural number, (n + 1) is 2n + 1.
Adding Triangular Numbers
Drawing number into triangular dot pattern and making a square (with same numbers of dots in rows and columns) will give a square number.
Math Patterns
Similarly, 16 = 42 = 32 + (3 + 3 + 1), 25 = 52 = 42 + (4 + 4 + 1) and so on.
Adding Odd Numbers
A number will be a square number if it is the sum of consecutive or successive odd numbers starting from 1. Let us now consider,
1 (first odd number) = 1 = 12
1 + 3 (sum of first two odd numbers) = 4 = 22
1 + 3 + 5 (sum of first three odd numbers) = 9 = 32
1 + 3 + 5 + 7 (sum of first four odd numbers) = 16 = 42
1 + 3 + 5 + 7 + 9 (sum of first five odd numbers) = 25 = 52
A number which is not a perfect square cannot be put in this form. Also, we can check if a number is a perfect square or not. Let us now consider the number 16. Subtracting 16 from the series of successive odd numbers and getting 0, as a result, will tell that 16 is a perfect square.
16 – 1 = 15
15 − 3 = 12
12 − 5 = 7
7 − 7 = 0